2002
2002
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1.I.1B
Part IA, 2002 comment(a) State the Orbit-Stabilizer Theorem for a finite group acting on a set .
(b) Suppose that is the group of rotational symmetries of a cube . Two regular tetrahedra and are inscribed in , each using half the vertices of . What is the order of the stabilizer in of ?
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1.I.2D
Part IA, 2002 commentState the Fundamental Theorem of Algebra. Define the characteristic equation for an arbitrary matrix whose entries are complex numbers. Explain why the matrix must have three eigenvalues, not necessarily distinct.
Find the characteristic equation of the matrix
and hence find the three eigenvalues of . Find a set of linearly independent eigenvectors, specifying which eigenvector belongs to which eigenvalue.
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1.II.5B
Part IA, 2002 comment(a) Find a subset of the Euclidean plane that is not fixed by any isometry (rigid motion) except the identity.
Let be a subgroup of the group of isometries of a subset of not fixed by any isometry except the identity, and let denote the union . Does the group of isometries of contain ? Justify your answer.
(b) Find an example of such a and with .
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1.II.6B
Part IA, 2002 comment(a) Suppose that is a Möbius transformation, acting on the extended complex plane. What are the possible numbers of fixed points that can have? Justify your answer.
(b) Show that the operation of complex conjugation, defined by , is not a Möbius transformation.
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1.II.7B
Part IA, 2002 comment(a) Find, with justification, the matrix, with respect to the standard basis of , of the rotation through an angle about the origin.
(b) Find the matrix, with respect to the standard basis of , of the rotation through an angle about the axis containing the point and the origin. You may express your answer in the form of a product of matrices.
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1.II.8D
Part IA, 2002 commentDefine what is meant by a vector space over the real numbers . Define subspace, proper subspace, spanning set, basis, and dimension.
Define the sum and intersection of two subspaces and of a vector space . Why is the intersection never empty?
Let and let , where , and let . Show that has the orthogonal basis where and . Extend this basis to find orthogonal bases of , and . Show that and hence verify that, in this case,
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3.I.1A
Part IA, 2002 commentGiven two real non-zero matrices and , with , show that maps onto a line. Is it always true that Show that there is always a non-zero matrix with . Justify your answers.
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3.I.2B
Part IA, 2002 comment(a) What does it mean for a group to be cyclic? Give an example of a finite abelian group that is not cyclic, and justify your assertion.
(b) Suppose that is a finite group of rotations of about the origin. Is necessarily cyclic? Justify your answer.
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3.II.5E
Part IA, 2002 commentProve, using the standard formula connecting and , that
Define, in terms of the dot and cross product, the triple scalar product [a, b, c of three vectors in and show that it is invariant under cyclic permutation of the vectors.
Let be a not necessarily orthonormal basis for , and define
By calculating , show that is also a basis for .
The vectors are constructed from in the same way that are constructed from . Show that
Show that a vector has components with respect to the basis . What are the components of the vector with respect to the basis ?
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3.II.7A
Part IA, 2002 commentExplain why the number of solutions of the simultaneous linear equations is 0,1 or infinite, where is a real matrix and . Let be the mapping which represents. State necessary and sufficient conditions on and for each of these possibilities to hold.
Let and be matrices representing linear mappings and . Give necessary and sufficient conditions on and for the existence of a matrix with . When is unique?
Find when
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3.II.8B
Part IA, 2002 commentSuppose that a,b, are the vertices of a regular tetrahedron in and that .
(a) Find and .
(b) Find a matrix that is a rotation leaving invariant such that and
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1.I
Part IA, 2002 commentSuppose for and . What does it mean to say that as ? What does it mean to say that as ?
Show that, if for all and as , then as . Is the converse true? Give a proof or a counter example.
Show that, if for all and with , then as .
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3.II.6E
Part IA, 2002 comment(a) Give the general solution for and of the equations
Show in particular that and must lie at opposite ends of a diameter of a sphere whose centre and radius should be specified.
(b) If two pairs of opposite edges of a tetrahedron are perpendicular, show that the third pair are also perpendicular to each other. Show also that the sum of the lengths squared of two opposite edges is the same for each pair.
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1.I.4C
Part IA, 2002 commentShow that any bounded sequence of real numbers has a convergent subsequence.
Give an example of a sequence of real numbers with no convergent subsequence.
Give an example of an unbounded sequence of real numbers with a convergent subsequence.
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1.II.9C
Part IA, 2002 commentState some version of the fundamental axiom of analysis. State the alternating series test and prove it from the fundamental axiom.
In each of the following cases state whether converges or diverges and prove your result. You may use any test for convergence provided you state it correctly.
(i) .
(ii) .
(iii) .
(iv) for .
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1.II.10C
Part IA, 2002 commentShow that a continuous real-valued function on a closed bounded interval is bounded and attains its bounds.
Write down examples of the following functions (no proof is required).
(i) A continuous function which is not bounded.
(ii) A continuous function which is bounded but does not attain its bounds.
(iii) A bounded function which is not continuous.
(iv) A function which is not bounded on any interval with
[Hint: Consider first how to define on the rationals.]
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1.II.11C
Part IA, 2002 commentState the mean value theorem and deduce it from Rolle's theorem.
Use the mean value theorem to show that, if is differentiable with for all , then is constant.
By considering the derivative of the function given by , find all the solutions of the differential equation where is differentiable and is a fixed real number.
Show that, if is continuous, then the function given by
is differentiable with .
Find the solution of the equation
where is differentiable and is a real number. You should explain why the solution is unique.
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1.II.12C
Part IA, 2002 commentProve Taylor's theorem with some form of remainder.
An infinitely differentiable function satisfies the differential equation
and the conditions . If and is a positive integer, explain why we can find an such that
for all with . Explain why we can find an such that
for all with and all .
Use your form of Taylor's theorem to show that
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2.I.1D
Part IA, 2002 commentSolve the equation
subject to the conditions at . Solve the equation
subject to the same conditions at .
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2.I.2D
Part IA, 2002 commentConsider the equation
where the positive square root is taken, within the square . Find the solution that begins at . Sketch the corresponding solution curve, commenting on how its tangent behaves near each extremity. By inspection of the righthand side of , or otherwise, roughly sketch, using small line segments, the directions of flow throughout the square .
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2.II.5D
Part IA, 2002 commentExplain what is meant by an integrating factor for an equation of the form
Show that is an integrating factor for
and find the solution such that , for given .
Show that for all and hence that
For a solution with , show graphically, by considering the sign of first for and then for , that for all .
Sketch the solution for the case , and show that property that both as and as from below, where is the positive number that satisfies .
[Do not consider the range .]
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2.II.6D
Part IA, 2002 commentSolve the differential equation
for the general initial condition at , where , and are positive constants. Deduce that the equilibria at and are stable and unstable, respectively.
By using the approximate finite-difference formula
for the derivative of at , where is a positive constant and , show that the differential equation when thus approximated becomes the difference equation
where and where . Find the two equilibria and, by linearizing the equation about them or otherwise, show that one is always unstable (given that ) and that the other is stable or unstable according as or . Show that this last instability is oscillatory with period . Why does this last instability have no counterpart for the differential equation? Show graphically how this instability can equilibrate to a periodic, finite-amplitude oscillation when .
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2.II.7D
Part IA, 2002 commentThe homogeneous equation
has non-constant, non-singular coefficients and . Two solutions of the equation, and , are given. The solutions are known to be such that the determinant
is non-zero for all . Define what is meant by linear dependence, and show that the two given solutions are linearly independent. Show also that
In the corresponding inhomogeneous equation
the right-hand side is a prescribed forcing function. Construct a particular integral of this inhomogeneous equation in the form
where the two functions are to be determined such that
for all . Express your result for the functions in terms of integrals of the functions and .
Consider the case in which for all and is a positive constant, say, and in which the forcing . Show that in this case and can be taken as and respectively. Evaluate and and show that, as , one of the increases in magnitude like a power of to be determined.
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2.II.8D
Part IA, 2002 commentFor any solution of the equations
show that
What does this imply about the behaviour of phase-plane trajectories at large distances from the origin as , in the case ? Give brief reasoning but do not try to find explicit solutions.
Analyse the properties of the critical points and sketch the phase portrait (a) in the case , (b) in the case , and (c) in the case .
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4.I.3E
Part IA, 2002 commentThe position of the leading edge of an avalanche moving down a mountain side making a positive angle to the horizontal satisfies the equation
where is the acceleration due to gravity.
By multiplying the equation by , obtain the first integral
where is an arbitrary constant of integration and the dot denotes differentiation with respect to time.
Sketch the positive quadrant of the phase plane. Show that all solutions approach the trajectory
Hence show that, independent of initial conditions, the avalanche ultimately has acceleration .
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4.I.4E
Part IA, 2002 commentAn inertial reference frame and another reference frame have a common origin O. rotates with constant angular velocity with respect to . Assuming the result that
for an arbitrary vector , show that
where is the position vector of a point measured from the origin.
A system of electrically charged particles, all with equal masses and charges , moves under the influence of mutual central forces of the form
In addition each particle experiences a Lorentz force due to a constant weak magnetic field given by
Transform the equations of motion to the rotating frame . Show that if the angular velocity is chosen to satisfy
and if terms of second order in are neglected, then the equations of motion in the rotating frame are identical to those in the non-rotating frame in the absence of the magnetic field B.
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4.II.9E
Part IA, 2002 commentWrite down the equations of motion for a system of gravitating point particles with masses and position vectors .
Assume that , where the vectors are independent of time . Obtain a system of equations for the vectors which does not involve the time variable .
Show that the constant vectors must be located at stationary points of the function
Show that for this system, the total angular momentum about the origin and the total momentum both vanish. What is the angular momentum about any other point?
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4.II.10E
Part IA, 2002 commentDerive the equation
for the orbit of a particle of mass and angular momentum moving under a central force directed towards a fixed point . Give an interpretation of in terms of the area swept out by a radius vector.
If the orbits are found to be circles passing through , then deduce that the force varies inversely as the fifth power of the distance, , where is a constant. Is the force attractive or repulsive?
Show that, for fixed mass, the radius of the circle varies inversely as the angular momentum of the particle, and hence that the time taken to traverse a complete circle is proportional to .
[You may assume, if you wish, the expressions for radial and transverse acceleration in the forms .]
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4.II.11E
Part IA, 2002 commentAn electron of mass moving with velocity in the vicinity of the North Pole experiences a force
where is a constant and the position vector of the particle is with respect to an origin located at the North Pole. Write down the equation of motion of the electron, neglecting gravity. By taking the dot product of the equation with show that the speed of the electron is constant. By taking the cross product of the equation with show that
where is a constant vector. By taking the dot product of this equation with , show that the electron moves on a cone centred on the North Pole.
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4.II.12E
Part IA, 2002 commentCalculate the moment of inertia of a uniform rod of length and mass about an axis through its centre and perpendicular to its length. Assuming it moves in a plane, give an expression for the kinetic energy of the rod in terms of the speed of the centre and the angle that it makes with a fixed direction.
Two such rods are freely hinged together at one end and the other two ends slide on a perfectly smooth horizontal floor. The rods are initially at rest and lie in a vertical plane, each making an angle to the horizontal. The rods subsequently move under gravity. Calculate the speed with which the hinge strikes the ground.
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4.I.1C
Part IA, 2002 commentWhat does it mean to say that a function is injective? What does it mean to say that a function is surjective?
Consider the functions and their composition given by . Prove the following results.
(i) If and are surjective, then so is .
(ii) If and are injective, then so is .
(iii) If is injective, then so is .
(iv) If is surjective, then so is .
Give an example where is injective and surjective but is not surjective and is not injective.
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4.I.2C
Part IA, 2002 commentIf are infinitely differentiable, Leibniz's rule states that, if ,
Prove this result by induction. (You should prove any results on binomial coefficients that you need.)
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4.II.5F
Part IA, 2002 commentWhat is meant by saying that a set is countable?
Prove that the union of countably many countable sets is itself countable.
Let be a collection of disjoint intervals of the real line, each having strictly positive length. Prove that the index set is countable.
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4.II.6F
Part IA, 2002 comment(a) Let be a finite set, and let be the power set of , that is, the set of all subsets of . Let be additive in the sense that whenever . Show that, for ,
(b) Let be finite sets. Deduce from part (a) the inclusion-exclusion formula for the size (or cardinality) of .
(c) A derangement of the set is a permutation (that is, a bijection from to itself) in which no member of the set is fixed (that is, for all ). Using the inclusion-exclusion formula, show that the number of derangements satisfies as .
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4.II
Part IA, 2002 comment(a) Suppose that is an odd prime. Find modulo .
(b) Find ! modulo , when is an odd prime.
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4.II.8B
Part IA, 2002 commentSuppose that are coprime positive integers. Write down an integer such that modulo . The least such is the order of modulo . Show that if the order of modulo is , and modulo , then divides .
Let and . Suppose that is a prime factor of . Find the order of 2 modulo , and show that modulo .
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2.I.3F
Part IA, 2002 commentDefine the indicator function of an event .
Let be the indicator function of the event , and let be the number of values of such that occurs. Show that where , and find in terms of the quantities .
Using Chebyshev's inequality or otherwise, show that
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2.I.4F
Part IA, 2002 commentA coin shows heads with probability on each toss. Let be the probability that the number of heads after tosses is even. Show carefully that , , and hence find . [The number 0 is even.]
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2.II.9F
Part IA, 2002 comment(a) Define the conditional probability of the event given the event . Let be a partition of the sample space such that for all . Show that, if ,
(b) There are urns, the th of which contains red balls and blue balls. You pick an urn (uniformly) at random and remove two balls without replacement. Find the probability that the first ball is blue, and the conditional probability that the second ball is blue given that the first is blue. [You may assume that .]
(c) What is meant by saying that two events and are independent?
(d) Two fair dice are rolled. Let be the event that the sum of the numbers shown is , and let be the event that the first die shows . For what values of and are the two events independent?
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2.II.10F
Part IA, 2002 commentThere is a random number of foreign objects in my soup, with mean and finite variance. Each object is a fly with probability , and otherwise is a spider; different objects have independent types. Let be the number of flies and the number of spiders.
(a) Show that denotes the probability generating function of a random variable . You should present a clear statement of any general result used.]
(b) Suppose has the Poisson distribution with parameter . Show that has the Poisson distribution with parameter , and that and are independent.
(c) Let and suppose that and are independent. [You are given nothing about the distribution of .] Show that . By working with the function or otherwise, deduce that has the Poisson distribution. [You may assume that as .]
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2.II.11F
Part IA, 2002 commentLet be independent random variables each with the uniform distribution on the interval .
(a) Show that has density function
(b) Show that .
(c) You are provided with three rods of respective lengths . Show that the probability that these rods may be used to form the sides of a triangle is .
(d) Find the density function of for . Let be uniformly distributed on , and independent of . Show that the probability that rods of lengths may be used to form the sides of a quadrilateral is .
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2.II.12F
Part IA, 2002 comment(a) Explain what is meant by the term 'branching process'.
(b) Let be the size of the th generation of a branching process in which each family size has probability generating function , and assume that . Show that the probability generating function of satisfies for .
(c) Show that is the probability generating function of a non-negative integer-valued random variable when , and find explicitly when is thus given.
(d) Find the probability that , and show that it converges as to . Explain carefully why this implies that the probability of ultimate extinction equals .
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3.I.3A
Part IA, 2002 commentDetermine whether each of the following is the exact differential of a function, and if so, find such a function: (a) , (b) .
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3.I.4A
Part IA, 2002 commentState the divergence theorem.
Consider the integral
where and is the sphere of radius centred at the origin. Evaluate directly, and by means of the divergence theorem.
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3.II.9A
Part IA, 2002 commentTwo independent variables and are related to a third variable by
where and are constants. Let be a smooth function of and , and let . Show, by using the Taylor series for about , that
where all derivatives are evaluated at .
Hence show that a stationary point of is a local minimum if
where is the Hessian matrix evaluated at .
Find two local minima of
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3.II.10A
Part IA, 2002 commentThe domain in the plane is bounded by and . Find a transformation
such that is transformed into a rectangle in the plane.
Evaluate
where is the region bounded by
and the planes
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3.II.11A
Part IA, 2002 commentProve that
is an open orientable surface in with unit normal , and is any continuously differentiable vector field such that on . Let be a continuously differentiable unit vector field which coincides with on . By applying Stokes' theorem to , show that
where denotes arc-length along the boundary of , and is such that . Verify this result by taking , and to be the disc in the plane.
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3.II.12A
Part IA, 2002 comment(a) Show, using Cartesian coordinates, that satisfies Laplace's equation, , on
(b) and are smooth functions defined in a 3-dimensional domain bounded by a smooth surface . Show that
(c) Let , and let be a domain bounded by a smooth outer surface and an inner surface , where is a sphere of radius , centre . The function satisfies
Use parts (a) and (b) to show, taking the limit , that at is given by
where is the domain bounded by .
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1.I.5G
Part IB, 2002 commentDefine by
Find the characteristic polynomial and the minimal polynomial of . Is diagonalisable? Are and linearly independent endomorphisms of ? Justify your answers.
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1.II.14G
Part IB, 2002 commentLet be an endomorphism of a vector space of finite dimension .
(a) What is the dimension of the vector space of linear endomorphisms of ? Show that there exists a non-trivial polynomial such that . Define what is meant by the minimal polynomial of .
(b) Show that the eigenvalues of are precisely the roots of the minimal polynomial of .
(c) Let be a subspace of such that and let be the restriction of to . Show that divides .
(d) Give an example of an endomorphism and a subspace as in (c) not equal to for which , and .
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2.I.6G
Part IB, 2002 commentLet be a complex matrix such that . What are the possible minimal polynomials of ? If is not diagonalisable and , list all possible Jordan normal forms of .
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2.II.15G
Part IB, 2002 comment(a) A complex matrix is said to be unipotent if is nilpotent, where is the identity matrix. Show that is unipotent if and only if 1 is the only eigenvalue of .
(b) Let be an invertible complex matrix. By considering the Jordan normal form of show that there exists an invertible matrix such that
where is an invertible diagonal matrix, is an upper triangular matrix with zeros in the diagonal and .
(c) Set and show that is unipotent.
(d) Conclude that any invertible matrix can be written as where is diagonalisable, is unipotent and .
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3.I
Part IB, 2002 commentWhich of the following statements are true, and which false? Give brief justifications for your answers.
(a) If and are subspaces of a vector space , then is always a subspace of .
(b) If and are distinct subspaces of a vector space , then is never a subspace of .
(c) If and are subspaces of a vector space , then .
(d) If is a subspace of a finite-dimensional space , then there exists a subspace such that and .
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3.II.17F
Part IB, 2002 commentDefine the determinant of an matrix , and prove from your definition that if is obtained from by an elementary row operation (i.e. by adding a scalar multiple of the th row of to the th row, for some ), then .
Prove also that if is a matrix of the form
where denotes the zero matrix, then det . Explain briefly how the matrix
can be transformed into the matrix
by a sequence of elementary row operations. Hence or otherwise prove that .
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4.I
Part IB, 2002 commentDefine the rank and nullity of a linear map between finite-dimensional vector spaces.
State the rank-nullity formula.
Let and be linear maps. Prove that
Part IB
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4.II.15F
Part IB, 2002 commentDefine the dual space of a finite-dimensional real vector space , and explain what is meant by the basis of dual to a given basis of . Explain also what is meant by the statement that the second dual is naturally isomorphic to .
Let denote the space of real polynomials of degree at most . Show that, for any real number , the function mapping to is an element of . Show also that, if are distinct real numbers, then is a basis of , and find the basis of dual to it.
Deduce that, for any distinct points of the interval , there exist scalars such that
for all . For and , find the corresponding scalars .
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1.I.4E
Part IB, 2002 commentShow that any finite group of orientation-preserving isometries of the Euclidean plane is cyclic.
Show that any finite group of orientation-preserving isometries of the hyperbolic plane is cyclic.
[You may assume that given any non-empty finite set in the hyperbolic plane, or the Euclidean plane, there is a unique smallest closed disc that contains E. You may also use any general fact about the hyperbolic plane without proof providing that it is stated carefully.]
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3.I.4E
Part IB, 2002 commentState Euler's formula for a graph with faces, edges and vertices on the surface of a sphere.
Suppose that every face in has at least three edges, and that at least three edges meet at every vertex of . Let be the number of faces of that have exactly edges , and let be the number of vertices at which exactly edges meet . By expressing in terms of the , or otherwise, show that every convex polyhedron has at least four faces each of which is a triangle, a quadrilateral or a pentagon.
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1.II.13E
Part IB, 2002 commentLet , and let have the hyperbolic metric derived from the line element . Let be the group of Möbius maps of the form , where and are real and . Show that every in is an isometry of the metric space . For and in , let
Show that for every in . By considering , where , and , or otherwise, show that for all and in ,
By considering points , where and , where , or otherwise, derive Pythagoras' Theorem for hyperbolic geometry in the form , where and are the lengths of sides of a right-angled triangle whose hypotenuse has length .
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3.II.14E
Part IB, 2002 commentShow that every isometry of Euclidean space is a composition of reflections in planes
What is the smallest integer such that every isometry of with can be expressed as the composition of at most reflections? Give an example of an isometry that needs this number of reflections and justify your answer.
Describe (geometrically) all twelve orientation-reversing isometries of a regular tetrahedron.
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1.I.1E
Part IB, 2002 commentSuppose that for each , the function is uniformly continuous on .
(a) If pointwise on is necessarily continuous on ?
(b) If uniformly on is necessarily continuous on ?
In each case, give a proof or a counter-example (with justification).
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1.II.10E
Part IB, 2002 commentSuppose that is a metric space that has the Bolzano-Weierstrass property (that is, any sequence has a convergent subsequence). Let be any metric space, and suppose that is a continuous map of onto . Show that also has the Bolzano-Weierstrass property.
Show also that if is a bijection of onto , then is continuous.
By considering the map defined on the real interval , or otherwise, show that there exists a continuous choice of arg for the complex number lying in the right half-plane .
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2.I.1E
Part IB, 2002 commentDefine what is meant by (i) a complete metric space, and (ii) a totally bounded metric space.
Give an example of a metric space that is complete but not totally bounded. Give an example of a metric space that is totally bounded but not complete.
Give an example of a continuous function that maps a complete metric space onto a metric space that is not complete. Give an example of a continuous function that maps a totally bounded metric space onto a metric space that is not totally bounded.
[You need not justify your examples.]
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2.II.10E
Part IB, 2002 comment(a) Let be a map of a complete metric space into itself, and suppose that there exists some in , and some positive integer , such that for all distinct and in , where is the th iterate of . Show that has a unique fixed point in .
(b) Let be a map of a compact metric space into itself such that for all distinct and in . By considering the function , or otherwise, show that has a unique fixed point in .
(c) Suppose that satisfies for every distinct and in . Suppose that for some , the orbit is bounded. Show that maps the closure of into itself, and deduce that has a unique fixed point in .
[The Contraction Mapping Theorem may be used without proof providing that it is correctly stated.]
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3.I.1E
Part IB, 2002 commentLet be defined by , where and are defined by and, for and . Show that is differentiable on .
Show that for any real non-zero , where we regard as the vector in .
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Part IB, 2002
commentShow that if and are non-negative numbers, and , then
Deduce that if is a metric space, then is a metric on .
Let and . Let be the class of continuous complex-valued functions on and, for and in , define
where . Show that the series for converges, and that is a metric on .
For , let and . Show that for . By considering the sums for and separately, show that for each ,
and deduce that as .
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4.I.1E
Part IB, 2002 comment(a) Let be a metric space containing the point , and let
Is necessarily the largest open subset of ? Is necessarily the smallest closed set that contains ? Justify your answers.
(b) Let be a normed space with norm , and let
Is necessarily the largest open subset of ? Is necessarily the smallest closed set that contains ? Justify your answers.
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4.II.10E
Part IB, 2002 comment(a) Let be a finite-dimensional real vector space, and let and be two norms on . Show that a function is differentiable at a point in with respect to if and only if it is differentiable at with respect to , and that if this is so then the derivative of is independent of the norm used. [You may assume that all norms on a finite-dimensional vector space are equivalent.]
(b) Let and be finite-dimensional normed real vector spaces with having norm , and let be a continuous bilinear mapping. Show that is differentiable at any point in , and that [You may assume that is a norm on , and that is compact.]
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1.I.7B
Part IB, 2002 commentUsing contour integration around a rectangle with vertices
prove that, for all real ,
Hence derive that the function is an eigenfunction of the Fourier transform
i.e. is a constant multiple of .
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1.II.16B
Part IB, 2002 comment(a) Show that if is an analytic function at and , then is conformal at , i.e. it preserves angles between paths passing through .
(b) Let be the disc given by , and let be the half-plane given by , where . Construct a map of the domain onto , and hence find a conformal mapping of onto the disc . [Hint: You may find it helpful to consider a mapping of the form , where ad .]
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2.I.7B
Part IB, 2002 commentSuppose that is analytic, and that is constant in an open disk . Use the Cauchy-Riemann equations to show that is constant in .
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2.II.16B
Part IB, 2002 commentA function has an isolated singularity at , with Laurent expansion
(a) Define res , the residue of at the point .
(b) Prove that if is a pole of order , then
(c) Using the residue theorem and the formula above show that
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4.I.8B
Part IB, 2002 commentLet be a function such that . Prove that
if and only if
where is the Fourier transform of .
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4.II.17B
Part IB, 2002 comment(a) Using the inequality for , show that, if is continuous for large , and if as , then
where .
(b) By integrating an appropriate function along the contour formed by the semicircles and in the upper half-plane with the segments of the real axis and , show that
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1.I.2A
Part IB, 2002 commentFind the Fourier sine series for , on . To which value does the series converge at ?
Now consider the corresponding cosine series for , on . Sketch the cosine series between and . To which value does the series converge at ? [You do not need to determine the cosine series explicitly.]
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1.II.11A
Part IB, 2002 commentThe potential , satisfies Laplace's equation everywhere except on a sphere of unit radius and as . The potential is continuous at , but the derivative of the potential satisfies
where is a constant. Use the method of separation of variables to find for both and .
[The Laplacian in spherical polar coordinates for axisymmetric systems is
You may assume that the equation
has polynomial solutions of degree , which are regular at , if and only if
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2.I.2C
Part IB, 2002 commentWrite down the transformation law for the components of a second-rank tensor explaining the meaning of the symbols that you use.
A tensor is said to have cubic symmetry if its components are unchanged by rotations of about each of the three co-ordinate axes. Find the most general secondrank tensor having cubic symmetry.
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2.II.11C
Part IB, 2002 commentIf is a vector, and
show for arbitrary scalars and that is a symmetric second-rank tensor.
Find the eigenvalues and eigenvectors of .
Suppose now that depends upon position and that . Find constants and such that
Hence or otherwise show that if vanishes everywhere on a surface that encloses a volume then
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3.I.2A
Part IB, 2002 commentWrite down the wave equation for the displacement of a stretched string with constant mass density and tension. Obtain the general solution in the form
where is the wave velocity. For a solution in the region , with and as , show that
is constant in time. Express in terms of the general solution in this case.
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3.II.12A
Part IB, 2002 commentConsider the real Sturm-Liouville problem
with the boundary conditions , where and are continuous and positive on . Show that, with suitable choices of inner product and normalisation, the eigenfunctions , form an orthonormal set.
Hence show that the corresponding Green's function satisfying
where is not an eigenvalue, is
where is the eigenvalue corresponding to .
Find the Green's function in the case where
with boundary conditions , and deduce, by suitable choice of , that
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4.I.2A
Part IB, 2002 commentUse the method of Lagrange multipliers to find the largest volume of a rectangular parallelepiped that can be inscribed in the ellipsoid
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4.II.11A
Part IB, 2002 commentA function is chosen to make the integral
stationary, subject to given values of and . Derive an analogue of the Euler-Lagrange equation for .
Solve this equation for the case where
in the interval and
as , whilst
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1.I.9D
Part IB, 2002 commentConsider a quantum mechanical particle of mass moving in one dimension, in a potential well
Sketch the ground state energy eigenfunction and show that its energy is , where satisfies
[Hint: You may assume that
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1.II.18D
Part IB, 2002 commentA quantum mechanical particle of mass moves in one dimension in the presence of a negative delta function potential
where is a parameter with dimensions of length.
(a) Write down the time-independent Schrödinger equation for energy eigenstates , with energy . By integrating this equation across , show that the gradient of the wavefunction jumps across according to
[You may assume that is continuous across ]
(b) Show that there exists a negative energy solution and calculate its energy.
(c) Consider a double delta function potential
For sufficiently small , this potential yields a negative energy solution of odd parity, i.e. . Show that its energy is given by
[You may again assume is continuous across .]
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2.I.9D
Part IB, 2002 commentFrom the expressions
show that
is an eigenfunction of and , and compute the corresponding eigenvalues.
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2.II.18D
Part IB, 2002 commentConsider a quantum mechanical particle moving in an upside-down harmonic oscillator potential. Its wavefunction evolves according to the time-dependent Schrödinger equation,
(a) Verify that
is a solution of equation (1), provided that
and
(b) Verify that provides a solution to (3), where is an arbitrary real constant.
(c) The expectation value of an operator at time is
where is the normalised wave function. Show that for given by (2),
Hence show that as ,
[Hint: You may use
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3.II.20D
Part IB, 2002 commentA quantum mechanical system has two states and , which are normalised energy eigenstates of a Hamiltonian , with
A general time-dependent state may be written
where and are complex numbers obeying .
(a) Write down the time-dependent Schrödinger equation for , and show that if the Hamiltonian is , then
For the general state given in equation (1) above, write down the probability to observe the system, at time , in a state , properly normalised so that .
(b) Now consider starting the system in the state at time , and evolving it with a different Hamiltonian , which acts on the states and as follows:
By solving the time-dependent Schrödinger equation for the Hamiltonian , find and in this case. Hence determine the shortest time such that is an eigenstate of with eigenvalue .
(c) Now consider taking the state from part (b), and evolving it for further length of time , with Hamiltonian , which acts on the states and as follows:
What is the final state of the system? Is this state observationally distinguishable from the original state ?
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3.I.10D
Part IB, 2002 commentWrite down the formulae for a Lorentz transformation with velocity taking one set of co-ordinates to another .
Imagine you observe a train travelling past Cambridge station at a relativistic speed . Someone standing still on the train throws a ball in the direction the train is moving, with speed . How fast do you observe the ball to be moving? Justify your answer.
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4.I D
Part IB, 2002 commentA particle with mass is observed to be at rest. It decays into a particle of mass , and a massless particle. Calculate the energies and momenta of both final particles.
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4.II.18D
Part IB, 2002 commentA javelin of length is thrown horizontally and lengthwise into a shed of length at a speed of , where is the speed of light.
(a) What is the length of the javelin in the rest frame of the shed?
(b) What is the length of the shed in the rest frame of the javelin?
(c) Draw a space-time diagram in the rest frame coordinates of the shed, showing the world lines of both ends of the javelin, and of the front and back of the shed. Draw a second space-time diagram in the rest frame coordinates of the javelin, again showing the world lines of both ends of the javelin and of the front and back of the shed.
(d) Clearly mark the space-time events corresponding to (A) the trailing end of the javelin entering the shed, and (B) the leading end of the javelin hitting the back of the shed. Give the corresponding and coordinates for both (A) and (B). Are these two events space-like, null or time-like separated? How does the javelin fit inside the shed, even though it is initially longer than the shed in its own rest frame?
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1.I.6C
Part IB, 2002 commentA fluid flow has velocity given in Cartesian co-ordinates as where is a constant and is time. Show that the flow is incompressible. Find a stream function and determine an equation for the streamlines at time .
At the points along the straight line segment are marked with dye. Show that at any later time the marked points continue to form a segment of a straight line. Determine the length of this line segment at time and the angle that it makes with the -axis.
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1.II.15C
Part IB, 2002 commentState the unsteady form of Bernoulli's theorem.
A spherical bubble having radius at time is located with its centre at the origin in unbounded fluid. The fluid is inviscid, has constant density and is everywhere at rest at . The pressure at large distances from the bubble has the constant value , and the pressure inside the bubble has the constant value . In consequence the bubble starts to collapse so that its radius at time is . Find the velocity everywhere in the fluid in terms of at time and, assuming that surface tension is negligible, show that satisfies the equation
Find the total kinetic energy of the fluid in terms of at time . Hence or otherwise obtain a first integral of the above equation.
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3.I.8C
Part IB, 2002 commentState and prove Kelvin's circulation theorem.
Consider a planar flow in the unbounded region outside a cylinder for which the vorticity vanishes everywhere at time . What may be deduced about the circulation around closed loops in the fluid at time :
(i) that do not enclose the cylinder;
(ii) that enclose the cylinder?
Give a brief justification for your answer in each case.
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3.II.18C
Part IB, 2002 commentUse Euler's equation to derive Bernoulli's theorem for the steady flow of an inviscid fluid of uniform density in the absence of body forces.
Such a fluid flows steadily through a long cylindrical elastic tube having circular cross-section. The variable measures distance downstream along the axis of the tube. The tube wall has thickness , so that if the external radius of the tube is , its internal radius is , where is a given slowly-varying function that tends to zero as . The elastic tube wall exerts a pressure on the fluid given as
where and are positive constants. Far upstream, has the constant value , the fluid pressure has the constant value , and the fluid velocity has the constant value . Assume that gravity is negligible and that varies sufficiently slowly that the velocity may be taken as uniform across the tube at each value of . Use mass conservation and Bernoulli's theorem to show that satisfies
Sketch a graph of against . Show that if exceeds a critical value , no such flow is possible and find .
Show that if everywhere, then for given the equation has two positive solutions for . Explain how the given value of determines which solution should be chosen.
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4.I
Part IB, 2002 commentIf is given in Cartesian co-ordinates as , with a constant, verify that
When incompressible fluid is placed in a stationary cylindrical container of radius with its axis vertical, the depth of the fluid is . Assuming that the free surface does not reach the bottom of the container, use cylindrical polar co-ordinates to find the equation of the free surface when the fluid and the container rotate steadily about this axis with angular velocity .
Deduce the angular velocity at which the free surface first touches the bottom of the container.
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4.II.16C
Part IB, 2002 commentUse Euler's equation to show that in a planar flow of an inviscid fluid the vorticity satisfies
Write down the velocity field associated with a point vortex of strength in unbounded fluid.
Consider now the flow generated in unbounded fluid by two point vortices of strengths and at and , respectively. Show that in the subsequent motion the quantity
remains constant. Show also that the separation of the vortices, , remains constant.
Suppose finally that and that the vortices are placed at time at positions and . What are the positions of the vortices at time ?
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2.I.5B
Part IB, 2002 commentApplying the Gram-Schmidt orthogonalization, compute a "skinny"
QR-factorization of the matrix
i.e. find a matrix with orthonormal columns and an upper triangular matrix such that .
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2.II.14B
Part IB, 2002 commentLet and let distinct points be given.
(a) Define the divided difference of order in terms of interpolating polynomials. Prove that it is a symmetric function of the variables .
(b) Prove the recurrence relation
(c) Hence or otherwise deduce that, for any , we have
(d) From the formulas above, show that, for any ,
where .
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3.I.6B
Part IB, 2002 commentFor numerical integration, a quadrature formula
is applied which is exact on , i.e., for all polynomials of degree .
Prove that such a formula is exact for all if and only if , are the zeros of an orthogonal polynomial which satisfies for all . [You may assume that has distinct zeros.]
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3.II.16B
Part IB, 2002 comment(a) Consider a system of linear equations with a non-singular square matrix . To determine its solution we apply the iterative method
Here , while the matrix is such that implies . The initial vector is arbitrary. Prove that, if the matrix possesses linearly independent eigenvectors whose corresponding eigenvalues satisfy , then the method converges for any choice of , i.e. as .
(b) Describe the Jacobi iteration method for solving . Show directly from the definition of the method that, if the matrix is strictly diagonally dominant by rows, i.e.
then the method converges.
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Part IB, 2002
commentState the factorization criterion for sufficient statistics and give its proof in the discrete case.
Let form a random sample from a Poisson distribution for which the value of the mean is unknown. Find a one-dimensional sufficient statistic for .
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1.II.12H
Part IB, 2002 commentSuppose we ask 50 men and 150 women whether they are early risers, late risers, or risers with no preference. The data are given in the following table.
Derive carefully a (generalized) likelihood ratio test of independence of classification. What is the result of applying this test at the level?
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Part IB, 2002
commentExplain what is meant by a uniformly most powerful test, its power function and size.
Let be independent identically distributed random variables with common density . Obtain the uniformly most powerful test of against alternatives and determine the power function of the test.
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2.II.12H
Part IB, 2002 commentFor ten steel ingots from a production process the following measures of hardness were obtained:
On the assumption that the variation is well described by a normal density function obtain an estimate of the process mean.
The manufacturer claims that he is supplying steel with mean hardness 75 . Derive carefully a (generalized) likelihood ratio test of this claim. Knowing that for the data above
what is the result of the test at the significance level?
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4.I.3H
Part IB, 2002 commentFrom each of 100 concrete mixes six sample blocks were taken and subjected to strength tests, the number out of the six blocks failing the test being recorded in the following table:
On the assumption that the probability of failure is the same for each block, obtain an unbiased estimate of this probability and explain how to find a confidence interval for it.
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4.II.12H
Part IB, 2002 commentExplain what is meant by a prior distribution, a posterior distribution, and a Bayes estimator. Relate the Bayes estimator to the posterior distribution for both quadratic and absolute error loss functions.
Suppose are independent identically distributed random variables from a distribution uniform on , and that the prior for is uniform on .
Calculate the posterior distribution for , given , and find the point estimate for under both quadratic and absolute error loss function.
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Part IB, 2002
commentConsider a two-person zero-sum game with a payoff matrix
where . Here, the entry of the matrix indicates the payoff to player one if he chooses move and player two move . Suppose player one chooses moves 1 and 2 with probabilities and . Write down the maximization problem for the optimal strategy and solve it for each value of .
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3.II.15H
Part IB, 2002 commentConsider the following linear programming problem
Write down the Phase One problem for (1) and solve it.
By using the solution of the Phase One problem as an initial basic feasible solution for the Phase Two simplex algorithm, solve (1), i.e., find the optimal tableau and read the optimal solution and optimal value from it.
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4.I.5H
Part IB, 2002 commentState and prove the max flow/min cut theorem. In your answer you should define clearly the following terms: flow, maximal flow, cut, capacity.
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4.II.14H
Part IB, 2002 commentA gambler at a horse race has an amount to bet. The gambler assesses , the probability that horse will win, and knows that has been bet on horse by others, for . The total amount bet on the race is shared out in proportion to the bets on the winning horse, and so the gambler's optimal strategy is to choose so that it maximizes
where is the amount the gambler bets on horse . Show that the optimal solution to (1) also solves the following problem:
Assume that . Applying the Lagrangian sufficiency theorem, prove that the optimal solution to (1) satisfies
with maximal possible .
[You may use the fact that for all , the minimum of the function on the non-negative axis is attained at
Deduce that if is small enough, the gambler's optimal strategy is to bet on the horses for which the ratio is maximal. What is his expected gain in this case?
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1.I.8F
Part IB, 2002 commentDefine the rank and signature of a symmetric bilinear form on a finite-dimensional real vector space. (If your definitions involve a matrix representation of , you should explain why they are independent of the choice of representing matrix.)
Let be the space of all real matrices (where ), and let be the bilinear form on defined by
Find the rank and signature of .
[Hint: You may find it helpful to consider the subspace of symmetric matrices having trace zero, and a suitable complement for this subspace.]
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1.II.17F
Part IB, 2002 commentLet and be real symmetric matrices, such that the quadratic form is positive definite. Show that it is possible to find an invertible matrix such that and is diagonal. Show also that the diagonal entries of the matrix may be calculated directly from and , without finding the matrix . If
find the diagonal entries of .
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2.I.8F
Part IB, 2002 commentExplain what is meant by a sesquilinear form on a complex vector space . If and are two such forms, and for all , prove that for all . Deduce that if is a linear map satisfying for all , then for all .
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2.II.17F
Part IB, 2002 commentDefine the adjoint of an endomorphism of a complex inner-product space . Show that if is a subspace of , then if and only if .
An endomorphism of a complex inner-product space is said to be normal if it commutes with its adjoint. Prove the following facts about a normal endomorphism of a finite-dimensional space .
(i) and have the same kernel.
(ii) and have the same eigenvectors, with complex conjugate eigenvalues.
(iii) If , then .
(iv) There is an orthonormal basis of consisting of eigenvectors of .
Deduce that an endomorphism is normal if and only if it can be written as a product , where is Hermitian, is unitary and and commute with each other. [Hint: Given , define and in terms of their effect on the basis constructed in (iv).]
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3.I
Part IB, 2002 commentExplain what is meant by a quadratic residue modulo an odd prime , and show that is a quadratic residue modulo if and only if . Hence characterize the odd primes for which is a quadratic residue.
State the law of quadratic reciprocity, and use it to determine whether 73 is a quadratic residue (mod 127).
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3.II.19F
Part IB, 2002 commentExplain what is meant by saying that a positive definite integral quadratic form is reduced, and show that every positive definite form is equivalent to a reduced form
State a criterion for a prime number to be representable by some form of discriminant , and deduce that is representable by a form of discriminant if and only if or . Find the reduced forms of discriminant , and hence or otherwise show that a prime is representable by the form if and only if .
[Standard results on when and 2 are squares (mod ) may be assumed.]
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2.I.4G
Part IB, 2002 commentLet the function be analytic in the complex plane with real-valued.
Prove that, if is bounded above everywhere on , then is constant.
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2.II.13G
Part IB, 2002 comment(a) Given a topology on , a collection is called a basis for if every non-empty set in is a union of sets in . Prove that a collection is a basis for some topology if it satisfies:
(i) the union of all sets in is ;
(ii) if for two sets and in , then there is a set with .
(b) On consider the dictionary order given by
if or if and . Given points and in let
Show that the sets for and in form a basis of a topology.
(c) Show that this topology on does not have a countable basis.
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3.I.3G
Part IB, 2002 commentLet be a continuous map between topological spaces. Let
(a) Show that if is Hausdorff, then is closed in .
(b) Show that if is compact, then is also compact.
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3.II.13G
Part IB, 2002 comment(a) Let and be two analytic functions on a domain and let be a simple closed curve homotopic in to a point. If for every in , prove that encloses the same number of zeros of as of .
(b) Let be an analytic function on the disk , for some . Suppose that maps the closed unit disk into the open unit disk (both centred at 0 ). Prove that has exactly one fixed point in the open unit disk.
(c) Prove that, if , then
has zeros in .
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4.I.4G
Part IB, 2002 comment(a) Let be a topological space and suppose , where and are disjoint nonempty open subsets of . Show that, if is a connected subset of , then is entirely contained in either or .
(b) Let be a topological space and let be a sequence of connected subsets of such that , for . Show that is connected.
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4.II.13G
Part IB, 2002 commentA function is said to be analytic at if there exists a real number such that is analytic for and is finite (i.e. has a removable singularity at . is said to have a pole at if has a pole at . Suppose that is a meromorphic function on the extended plane , that is, is analytic at each point of except for poles.
(a) Show that if has a pole at , then there exists such that has no poles for .
(b) Show that the number of poles of is finite.
(c) By considering the Laurent expansions around the poles show that is in fact a rational function, i.e. of the form , where and are polynomials.
(d) Deduce that the only bijective meromorphic maps of onto itself are the Möbius maps.
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A2.7
Part II, 2002 comment(i)
Consider the surface
where is a term of order at least 3 in . Calculate the first fundamental form at .
(ii) Calculate the second fundamental form, at , of the surface given in Part (i). Calculate the Gaussian curvature. Explain why your answer is consistent with Gauss' "Theorema Egregium".
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A3.7
Part II, 2002 comment(i) State what it means for surfaces and to be isometric.
Let be a surface, a diffeomorphism, and let
State a formula comparing the first fundamental forms of and .
(ii) Give a proof of the formula referred to at the end of part (i). Deduce that "isometry" is an equivalence relation.
The catenoid and the helicoid are the surfaces defined by
and
Show that the catenoid and the helicoid are isometric.
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A
Part II, 2002 commentWrite an essay on the Euler number of topological surfaces. Your essay should include a definition of subdivision, some examples of surfaces and their Euler numbers, and a discussion of the statement and significance of the Gauss-Bonnet theorem.
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A1.8
Part II, 2002 comment(i) State and prove a necessary and sufficient condition for a graph to be Eulerian (that is, to have an Eulerian circuit).
Prove that, given any connected non-Eulerian graph , there is an Eulerian graph and a vertex such that .
(ii) Let be a connected plane graph with vertices, edges and faces. Prove that . Deduce that , where is the smallest face size.
The crossing number of a non-planar graph is the minimum number of edgecrossings needed when drawing the graph in the plane. (The crossing of three edges at the same point is not allowed.) Show that if has vertices and edges then . Find .
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A2.8
Part II, 2002 comment(i) Define the chromatic polynomial of the graph , and establish the standard identity
where is an edge of . Deduce that, if has vertices and edges, then
where and for .
(ii) Let and be as in Part (i). Show that if has components then . Deduce that and for .
Show that if is a tree then . Must the converse hold? Justify your answer.
Show that if , where is a Turán graph, then .
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A4.9
Part II, 2002 commentWrite an essay on connectivity in graphs.
Your essay should include proofs of at least two major theorems, along with a discussion of one or two significant corollaries.
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A1.9
Part II, 2002 comment(i) Let be a prime number. Prove that the multiplicative group of the field with elements is cyclic.
(ii) Let be an odd prime, and let be an integer. Prove that we have if and only if either or . Is this statement true when ?
Let be an odd positive integer, and let be the number of distinct prime factors of . Prove that there are precisely different integers satisfying and .
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A3.9
Part II, 2002 comment(i) Let denote the number of primes , where is a positive real number. State and prove Legendre's formula relating to . Use this formula to compute
(ii) Let , where is a real number greater than 1 . Prove the following two assertions rigorously, assuming always that . (a) , where the product is taken over all primes ; (b) .
Explain why (b) enables us to define for . Deduce from (b) that .
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A4.10
Part II, 2002 commentWrite an essay on quadratic reciprocity. Your essay should include (i) a proof of the law of quadratic reciprocity for the Legendre symbol, (ii) a proof of the law of quadratic reciprocity for the Jacobi symbol, and (iii) a comment on why this latter law is useful in primality testing.
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A1.10
Part II, 2002 comment(i) Describe the original Hamming code of length 7 . Show how to encode a message word, and how to decode a received word involving at most one error. Explain why the procedure works.
(ii) What is a linear binary code? What is its dual code? What is a cyclic binary code? Explain how cyclic binary codes of length correspond to polynomials in dividing . Show that the dual of a cyclic code of length is cyclic of length .
Using the factorization
in , find all cyclic binary codes of length 7 . Identify those which are Hamming codes and their duals. Justify your answer.
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A2.9
Part II, 2002 comment(i) Explain the idea of public key cryptography. Give an example of a public key system, explaining how it works.
(ii) What is a general feedback register of length with initial fill ? What is the maximal period of such a register, and why? What does it mean for such a register to be linear?
Describe and justify the Berlekamp-Massey algorithm for breaking a cypher stream arising from a general linear feedback register of unknown length.
Use the Berlekamp-Massey algorithm to find a linear recurrence in with first eight terms .
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A2.10
Part II, 2002 comment(i) Let be a directed network with nodes , arcs and capacities specified on each of the arcs. Define the terms feasible flow, divergence, cut, upper and lower cut capacities. Given two disjoint sets of nodes and , what does it mean to say that a cut separates from ? Prove that the flux of a feasible flow from to is bounded above by the upper capacity of , for any cut separating from .
(ii) Define the maximum-flow and minimum-cut problems. State the max-flow min-cut theorem and outline the main steps of the maximum-flow algorithm. Use the algorithm to find the maximum flow between the nodes 1 and 5 in a network whose node set is , where the lower capacity of each arc is 0 and the upper capacity of the directed arc joining node to node is given by the -entry in the matrix
[The painted-network theorem can be used without proof but should be stated clearly. You may assume in your description of the maximum-flow algorithm that you are given an initial feasible flow.]
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A3.10
Part II, 2002 comment(i) Consider the unconstrained geometric programme GP
subject to .
State the dual problem to GP. Give a careful statement of the AM-GM inequality, and use it to prove the primal-dual inequality for GP.
(ii) Define min-path and max-tension problems. State and outline the proof of the max-tension min-path theorem.
A company has branches in five cities and . The fares for direct flights between these cities are as follows:
\begin{tabular}{l|lllll} & & & & & \ \hline & & 50 & 40 & 25 & 10 \ & 50 & & 20 & 90 & 25 \ & 40 & 20 & & 10 & 25 \ & 25 & 90 & 10 & & 55 \ & 10 & 25 & 25 & 55 & \end{tabular}
Formulate this as a min-path problem. Illustrate the max-tension min-path algorithm by finding the cost of travelling by the cheapest routes between and each of the other cities.
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A4.11
Part II, 2002 commentWrite an essay on Strong Lagrangian problems. You should give an account of duality and how it relates to the Strong Lagrangian property. In particular, establish carefully the relationship between the Strong Lagrangian property and supporting hyperplanes.
Also, give an example of a class of problems that are Strong Lagrangian. [You should explain carefully why your example has the Strong Lagrangian property.]
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A1.13
Part II, 2002 comment(i) Suppose are independent Poisson variables, and
where are unknown parameters, and are given covariates, each of dimension . Obtain the maximum-likelihood equations for , and explain briefly how you would check the validity of this model.
(ii) The data below show , which are the monthly accident counts on a major US highway for each of the 12 months of 1970 , then for each of the 12 months of 1971 , and finally for the first 9 months of 1972 . The data-set is followed by the (slightly edited) output. You may assume that the factors 'Year' and 'month' have been set up in the appropriate fashion. Give a careful interpretation of this output, and explain (a) how you would derive the corresponding standardised residuals, and (b) how you would predict the number of accidents in October 1972 .
first.glm Year month, poisson summary(first.glm
Call:
formula Year month, family poisson
\begin{tabular}{lrlll} Coefficients: & & & & \ (Intercept) & Estimate & Std. Error & \multicolumn{1}{l}{ z value } & \ Year1971 & & & & \ Year1972 & & & & \ month2 & & & & \ month3 & & & & \ month4 & & & & \ month5 & & & & \ month6 & & & & \ month7 & & & & \ month8 & & & & \ month9 & & & & \ month10 & & & & \ month11 & & & & \ month12 & & & & \end{tabular}
Signif. codes: 0 (, (, (, '.
(Dispersion parameter for poisson family taken to be 1 )
Number of Fisher Scoring iterations: 3
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A2.12
Part II, 2002 comment(i) Suppose that the random variable has density function of the form
where . Show that has expectation and variance .
(ii) Suppose now that are independent negative exponential variables, with having density function for . Suppose further that for , where is a known 'link' function, and are given covariate vectors, each of dimension . Discuss carefully the problem of finding , the maximum-likelihood estimator of , firstly for the case , and secondly for the case ; in both cases you should state the large-sample distribution of .
[Any standard theorems used need not be proved.]
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A4.14
Part II, 2002 commentAssume that the -dimensional observation vector may be written as , where is a given matrix of rank is an unknown vector, with , and
where is unknown. Find , the least-squares estimator of , and describe (without proof) how you would test
for a given .
Indicate briefly two plots that you could use as a check of the assumption .
Continued opposite Sulphur dioxide is one of the major air pollutants. A data-set presented by Sokal and Rohlf (1981) was collected on 41 US cities in 1969-71, corresponding to the following variables:
sulphur dioxide content of air in micrograms per cubic metre
average annual temperature in degrees Fahrenheit
= number of manufacturing enterprises employing 20 or more workers
population size (1970 census) in thousands
average annual wind speed in miles per hour
average annual precipitation in inches
average annual of days with precipitation per year
Interpret the output that follows below, quoting any standard theorems that you need to use.
Residuals :
Signif. codes: 0 ', ', ', ':
Residual standard error: on 34 degrees of freedom
Multiple R-Squared:
F-statistic: on 6 and 34 degrees of freedom, p-value:
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A1.14
Part II, 2002 comment(i) A system of identical non-interacting bosons has energy levels with degeneracy , for each particle. Show that in thermal equilibrium the number of particles with energy is given by
where and are parameters whose physical significance should be briefly explained.
(ii) A photon moves in a cubical box of side . Assuming periodic boundary conditions, show that, for large , the number of photon states lying in the frequency range is where
If the box is filled with thermal radiation at temperature , show that the number of photons per unit volume in the frequency range is where
Calculate the energy density of the thermal radiation. Show that the pressure exerted on the surface of the box satisfies
[You may use the result .]
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A2.14
Part II, 2002 comment(i) A simple model of a one-dimensional crystal consists of an infinite array of sites equally spaced with separation . An electron occupies the th site with a probability amplitude . The time-dependent Schrödinger equation governing these amplitudes is
where is the energy of an electron at an isolated site and the amplitude for transition between neighbouring sites is . By examining a solution of the form
show that , the energy of the electron in the crystal, lies in a band
Identify the Brillouin zone for this model and explain its significance.
(ii) In the above model the electron is now subject to an electric field in the direction of increasing . Given that the charge on the electron is write down the norm of the time-dependent Schrödinger equation for the probability amplitudes. Show that it has a solution of the form
where
Explain briefly how to interpret this result and use it to show that the dynamical behaviour of an electron near the bottom of the energy band is the same as that for a free particle in the presence of an electric field with an effective mass .
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A4.16
Part II, 2002 commentExplain how the energy band structure for electrons determines the conductivity properties of crystalline materials.
A semiconductor has a conduction band with a lower edge and a valence band with an upper edge . Assuming that the density of states for electrons in the conduction band is
and in the valence band is
where and are constants characteristic of the semiconductor, explain why at low temperatures the chemical potential for electrons lies close to the mid-point of the gap between the two bands.
Describe what is meant by the doping of a semiconductor and explain the distinction between -type and -type semiconductors, and discuss the low temperature limit of the chemical potential in both cases. Show that, whatever the degree and type of doping,
where is the density of electrons in the conduction band and is the density of holes in the valence band.
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A1.16
Part II, 2002 comment(i) Consider a one-dimensional model universe with "stars" distributed at random on the -axis, and choose the origin to coincide with one of the stars; call this star the "homestar." Home-star astronomers have discovered that all other stars are receding from them with a velocity , that depends on the position of the star. Assuming non-relativistic addition of velocities, show how the assumption of homogeneity implies that for some constant .
In attempting to understand the history of their one-dimensional universe, homestar astronomers seek to determine the velocity at time of a star at position . Assuming homogeneity, show how is determined in terms of a scale factor and hence deduce that for some function . What is the relation between and ?
(ii) Consider a three-dimensional homogeneous and isotropic universe with mass density , pressure and scale factor . Given that is the energy in volume , show how the relation yields the "fluid" equation
where .
Show how conservation of energy applied to a test particle at the boundary of a spherical fluid element yields the Friedmann equation
for constant . Hence obtain an equation for the acceleration in terms of and .
A model universe has mass density and pressure
where is constant. What does the fluid equation imply about ? Show that the acceleration vanishes if
Hence show that this universe is static and determine the sign of the constant .
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A3.14
Part II, 2002 comment(i) Write down the first law of thermodynamics for the change in the internal energy of a gas of particles in a volume with entropy .
Given that
where is the pressure, use the first law to show that is constant at constant and
Write down the Boyle-Charles law for a non-relativistic ideal gas and hence deduce that the temperature is proportional to at constant and .
State the principle of equipartition of energy and use it to deduce that
Hence deduce the value of . Show that this value of is such that the ratio is unchanged by a change of volume at constant and , where is the energy of the -th one particle eigenstate of a non-relativistic ideal gas.
(ii) A classical gas of non-relativistic particles of mass at absolute temperature and number density has a chemical potential
where is the particle's spin degeneracy factor. What condition on is needed for the validity of this formula and why?
Thermal and chemical equilibrium between two species of non-relativistic particles and is maintained by the reaction
where and are massless particles with zero chemical potential. Given that particles and have masses and respectively, but equal spin degeneracy factors, find the number density ratio as a function of and . Given that but show that
for some function which you should determine.
Explain how a reaction of the above type is relevant to a determination of the neutron to proton ratio in the early universe and why this ratio does not fall rapidly to zero as the universe cools. Explain briefly the process of primordial nucleosynthesis by which neutrons are converted into stable helium nuclei. Let
be the fraction of the universe that ends up in helium. Compute as a function of the ratio at the time of nucleosynthesis.
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A4.18
Part II, 2002 commentWhat is an ideal gas? Explain how the microstates of an ideal gas of indistinguishable particles can be labelled by a set of integers. What range of values do these integers take for (a) a boson gas and (b) a Fermi gas?
Let be the energy of the -th one-particle energy eigenstate of an ideal gas in thermal equilibrium at temperature and let be the probability that there are particles of the gas in this state. Given that
determine the normalization factor for (a) a boson gas and (b) a Fermi gas. Hence obtain an expression for , the average number of particles in the -th one-particle energy eigenstate for both cases (a) and (b).
In the case of a Fermi gas, write down (without proof) the generalization of your formula for to a gas at non-zero chemical potential . Show how it leads to the concept of a Fermi energy for a gas at zero temperature. How is related to the Fermi momentum for (a) a non-relativistic gas and (b) an ultra-relativistic gas?
In an approximation in which the discrete set of energies is replaced with a continuous set with momentum , the density of one-particle states with momentum in the range to is . Explain briefly why
where is the volume of the gas. Using this formula, obtain an expression for the total energy of an ultra-relativistic gas at zero chemical potential as an integral over . Hence show that
where is a number that you should compute. Why does this result apply to a photon gas?
Using the formula for a non-relativistic Fermi gas at zero temperature, obtain an expression for the particle number density in terms of the Fermi momentum and provide a physical interpretation of this formula in terms of the typical de Broglie wavelength. Obtain an analogous formula for the (internal) energy density and hence show that the pressure behaves as
where is a number that you should compute. [You need not prove any relation between the pressure and the energy density you use.] What is the origin of this pressure given that by assumption? Explain briefly and qualitatively how it is relevant to the stability of white dwarf stars.
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A1.17
Part II, 2002 comment(i) Let be a normal subgroup of the group . Let denote the group of cosets for . If is a representation of with for all show that is well-defined and that it is a representation of . Show further that is irreducible if and only if is irreducible.
(ii) For a matrix define the linear map by with as the vector of the Pauli spin matrices
Show that . Because of the linearity of there exists a matrix such that . Given that any matrix can be written as
where and is a unit vector, deduce that for all . Compute and in the case that and deduce that is the matrix of a rotation about with angle .
[Hint:
Show that defines a surjective homomorphism and find the kernel of .
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A3.15
Part II, 2002 comment(i) Let denote the symmetry group of rotations and reflections of a regular hexagon. The elements of are given by with and . The conjugacy classes of are and .
Show that the character table of is
\begin{tabular}{l|rrrrrr} & & & & & & \ \hline & 1 & 1 & 1 & 1 & 1 & 1 \ & 1 & 1 & 1 & 1 & & \ & 1 & & 1 & & 1 & \ & 1 & & 1 & & & 1 \ & 2 & 1 & & & 0 & 0 \ & 2 & & & 2 & 0 & 0 \end{tabular}
(ii) Show that the character of an rotation with angle in the dimensional irreducible representation of is given by
For a hexagonal crystal of atoms find how the degeneracy of the D-wave orbital states in the atomic central potential is split by the crystal potential with symmetry and give the new degeneracies.
By using the fact that is isomorphic to , or otherwise, find the degeneracies of eigenstates if the hexagonal symmetry is broken to the subgroup by a deformation. The introduction of a magnetic field further reduces the symmetry to . What will the degeneracies of the energy eigenstates be now?
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A1.18
Part II, 2002 comment(i) Material of thermal diffusivity occupies the semi-infinite region and is initially at uniform temperature . For time the temperature at is held at a constant value . Given that the temperature in satisfies the diffusion equation , write down the equation and the boundary and initial conditions satisfied by the dimensionless temperature .
Use dimensional analysis to show that the lengthscale of the region in which is significantly different from is proportional to . Hence show that this problem has a similarity solution
where .
What is the rate of heat input, , across the plane
(ii) Consider the same problem as in Part (i) except that the boundary condition at is replaced by one of constant rate of heat input . Show that satisfies the partial differential equation
and write down the boundary conditions on . Deduce that the problem has a similarity solution of the form
Derive the ordinary differential equation and boundary conditions satisfied by .
Differentiate this equation once to obtain
and solve for . Hence show that
Sketch the temperature distribution for various times , and calculate explicitly.
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A3.16
Part II, 2002 comment(i) A layer of fluid of depth , density and viscosity sits on top of a rigid horizontal plane at . Gravity acts vertically and surface tension is negligible.
Assuming that the horizontal velocity component and pressure satisfy the lubrication equations
together with appropriate boundary conditions at and (which should be stated), show that satisfies the partial differential equation
where .
(ii) A two-dimensional blob of the above fluid has fixed area and time-varying width , such that
The blob spreads under gravity.
Use scaling arguments to show that, after an initial transient, is proportional to and is proportional to . Hence show that equation of Part (i) has a similarity solution of the form
and find the differential equation satisfied by .
Deduce that
where
Express in terms of the integral
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A4.19
Part II, 2002 comment(a) A biological vessel is modelled two-dimensionally as a fluid-filled channel bounded by parallel plane walls , embedded in an infinite region of fluid-saturated tissue. In the tissue a solute has concentration , diffuses with diffusivity and is consumed by biological activity at a rate per unit volume, where and are constants. By considering the solute balance in a slice of tissue of infinitesimal thickness, show that
A steady concentration profile results from a flux , per unit area of wall, of solute from the channel into the tissue, where is a constant concentration of solute that is maintained in the channel and . Write down the boundary conditions satisfied by . Solve for and show that
where .
(b) Now let the solute be supplied by steady flow down the channel from one end, , with the channel taken to be semi-infinite in the -direction. The cross-sectionally averaged velocity in the channel varies due to a flux of fluid from the tissue to the channel (by osmosis) equal to per unit area. Neglect both the variation of across the channel and diffusion in the -direction.
By considering conservation of fluid, show that
and write down the corresponding equation derived from conservation of solute. Deduce that
where and .
Assuming that equation still holds, even though is now a function of as well as , show that satisfies the ordinary differential equation
Find scales and such that the dimensionless variables and satisfy
Derive the solution and find the constant .
To what values do and tend as ?
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A1.19
Part II, 2002 comment(i) In a reference frame rotating about a vertical axis with constant angular velocity the horizontal components of the momentum equation for a shallow layer of inviscid, incompressible fluid of constant density are
where and are independent of the vertical coordinate .
Define the Rossby number for a flow with typical velocity and lengthscale . What is the approximate form of the above equations when ?
Show that the solution to the approximate equations is given by a streamfunction proportional to .
Conservation of potential vorticity for such a flow is represented by
where is the vertical component of relative vorticity and is the thickness of the layer. Explain briefly why the potential vorticity of a column of fluid should be conserved.
(ii) Suppose that the thickness of the rotating, shallow-layer flow in Part (i) is where and are constants. By linearising the equation of conservation of potential vorticity about , show that the stream function for small disturbances to the state of rest obeys
where is a constant that should be found.
Obtain the dispersion relationship for plane-wave solutions of the form . Hence calculate the group velocity.
Show that if then the phase of these waves always propagates to the left (negative direction) but that the energy may propagate to either left or right.
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A2.16
Part II, 2002 comment(i) State the equations that relate strain to displacement and stress to strain in a linear, isotropic elastic solid.
In the absence of body forces, the Euler equation for infinitesimal deformations of a solid of density is
Derive an equation for in a linear, isotropic, homogeneous elastic solid. Hence show that both the dilatation and the rotation satisfy wave equations and find the corresponding wave speeds and .
(ii) The ray parameter is constant along seismic rays in a spherically symmetric Earth, where is the relevant wave speed or and is the angle between the ray and the local radial direction.
Express and sec in terms of and the variable . Hence show that the angular distance and travel time between a surface source and receiver, both at radius , are given by
where is the minimum radius attained by the ray. What is ?
A simple Earth model has a solid mantle in and a liquid core in . If in the mantle, where is a constant, find and for -arrivals (direct paths lying entirely in the mantle), and show that
[You may assume that .]
Sketch the curves for and arrivals on the same diagram and explain briefly why they terminate at .
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A4.20
Part II, 2002 commentThe equation of motion for small displacements in a homogeneous, isotropic, elastic material is
where and are the Lamé constants. Derive the conditions satisfied by the polarisation and (real) vector slowness s of plane-wave solutions , where is an arbitrary scalar function. Describe the division of these waves into -waves, -waves and -waves.
A plane harmonic -wave of the form
travelling through homogeneous elastic material of -wave speed and -wave speed is incident from on the boundary of rigid material in in which the displacement is identically zero.
Write down the form of the reflected wavefield in . Calculate the amplitudes of the reflected waves in terms of the components of the slowness vectors.
Derive expressions for the components of the incident and reflected slowness vectors, in terms of the wavespeeds and the angle of incidence . Hence show that there is no reflected -wave if
Sketch the rays produced if the region is fluid instead of rigid.
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A2.17
Part II, 2002 comment(i) Show that the equation
has roots in the neighbourhood of and . Find the first two terms of an expansion in for each of these roots.
Find a suitable series expansion for the other two roots and calculate the first two terms in each case.
(ii) Describe, giving reasons for the steps taken, how the leading-order approximation for to an integral of the form
where and are real, may be found by the method of stationary phase. Consider the cases where (a) has one simple zero at with ; (b) has more than one simple zero in ; and (c) has only a simple zero at . What is the order of magnitude of if is non-zero for ?
Use the method of stationary phase to find the leading-order approximation to
for .
[You may use the fact that .]
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A3.17
Part II, 2002 comment(i) State the Fredholm alternative for Fredholm integral equations of the second kind.
Show that the integral equation
where is a continuous function, has a unique solution for if . Derive this solution.
(ii) Describe the WKB method for finding approximate solutions of the equation
where is an arbitrary non-zero, differentiable function and is a small parameter. Obtain these solutions in terms of an exponential with slowly varying exponent and slowly varying amplitude.
Hence, by means of a suitable change of independent variable, find approximate solutions of the equation
in , where is a large parameter.
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A4.21
Part II, 2002 commentState Watson's lemma giving an asymptotic expansion as for an integral of the form
Show how this result may be used to find an asymptotic expansion as for an integral of the form
Hence derive Laplace's method for obtaining an asymptotic expansion as for an integral of the form
where is differentiable, for the cases: (i) in ; and (ii) has a simple zero at with and .
Find the first two terms in the asymptotic expansion as of
[You may leave your answer expressed in terms of -functions.]
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A2.18
Part II, 2002 comment(i) Find a travelling wave solution of unchanging shape for the modified Burgers equation (with )
with far ahead of the wave and far behind. What is the velocity of the wave? Sketch the shape of the wave.
(ii) Explain why the method of characteristics, when applied to an equation of the type
with initial data , sometimes gives a multi-valued solution. State the shockfitting algorithm that gives a single-valued solution, and explain how it is justified.
Consider the equation above, with . Suppose that
Sketch the characteristics in the plane. Show that a shock forms immediately, and calculate the velocity at which it moves.
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A3.18
Part II, 2002 comment(i) Show that the equation
has two solutions which are independent of both and . Show that one of these is linearly stable. Show that the other solution is linearly unstable, and find the range of wavenumbers that exhibit the instability.
Sketch the nonlinear evolution of the unstable solution after it receives a small, smooth, localized perturbation in the direction towards the stable solution.
(ii) Show that the equations
are a Bäcklund pair for the equations
By choosing to be a suitable constant, and using the Bäcklund pair, find a solution of the equation
which is non-singular in the region of the plane and has the value at .
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A1.1 B1.1
Part II, 2002 comment(i) We are given a finite set of airports. Assume that between any two airports, and , there are flights in each direction on every day. A confused traveller takes one flight per day, choosing at random from all available flights. Starting from , how many days on average will pass until the traveller returns again to ? Be careful to allow for the case where there may be no flights at all between two given airports.
(ii) Consider the infinite tree with root , where, for all , all vertices at distance from have degree 3 , and where all other vertices (except ) have degree 2 . Show that the random walk on is recurrent.
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A2.1
Part II, 2002 comment(i) In each of the following cases, the state-space and non-zero transition rates of a continuous-time Markov chain are given. Determine in which cases the chain is explosive.
(ii) Children arrive at a see-saw according to a Poisson process of rate 1 . Initially there are no children. The first child to arrive waits at the see-saw. When the second child arrives, they play on the see-saw. When the third child arrives, they all decide to go and play on the merry-go-round. The cycle then repeats. Show that the number of children at the see-saw evolves as a Markov Chain and determine its generator matrix. Find the probability that there are no children at the see-saw at time .
Hence obtain the identity
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A3.1 B3.1
Part II, 2002 comment(i) Consider the continuous-time Markov chain on with generator matrix
Compute the probability, starting from state 3 , that hits state 2 eventually.
Deduce that
[Justification of standard arguments is not expected.]
(ii) A colony of cells contains immature and mature cells. Each immature cell, after an exponential time of parameter 2, becomes a mature cell. Each mature cell, after an exponential time of parameter 3, divides into two immature cells. Suppose we begin with one immature cell and let denote the expected number of immature cells at time . Show that
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A4.1
Part II, 2002 commentWrite an essay on the long-time behaviour of discrete-time Markov chains on a finite state space. Your essay should include discussion of the convergence of probabilities as well as almost-sure behaviour. You should also explain what happens when the chain is not irreducible.
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A1.2 B1.2
Part II, 2002 comment(i) Derive Hamilton's equations from Lagrange's equations. Show that the Hamiltonian is constant if the Lagrangian does not depend explicitly on time.
(ii) A particle of mass is constrained to move under gravity, which acts in the negative -direction, on the spheroidal surface , with . If parametrize the surface so that
find the Hamiltonian .
Show that the energy
is a constant of the motion, where is a non-negative constant.
Rewrite this equation as
and sketch for and , identifying the maximal and minimal values of for fixed and . If is now taken not to be unity, how do these values depend on ?
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A2.2 B2.1
Part II, 2002 comment(i) A number of non-interacting particles move in one dimension in a potential . Write down the Hamiltonian and Hamilton's equations for one particle.
At time , the number density of particles in phase space is . Write down the time derivative of along a particle's trajectory. By equating the rate of change of the number of particles in a fixed domain in phase space to the flux into across its boundary, deduce that is a constant along any particle's trajectory.
(ii) Suppose that , and particles are injected in such a manner that the phase space density is a constant at any point of phase space corresponding to a particle energy being smaller than and zero elsewhere. How many particles are present?
Suppose now that the potential is very slowly altered to the square well form
Show that the greatest particle energy is now
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A3.2
Part II, 2002 comment(i) Show that Hamilton's equations follow from the variational principle
under the restrictions . Comment on the difference from the variational principle for Lagrange's equations.
(ii) Suppose we transform from and to and , with
where is the new Hamiltonian. Show that and obey Hamilton's equations with Hamiltonian .
Show that the time independent generating function takes the Hamiltonian
to harmonic oscillator form. Show that and obey the Poisson bracket relation
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A4.2
Part II, 2002 commentExplain how the orientation of a rigid body can be specified by means of the three Eulerian angles, and .
An axisymmetric top of mass has principal moments of inertia and , and is spinning with angular speed about its axis of symmetry. Its centre of mass lies a distance from the fixed point of support. Initially the axis of symmetry points vertically upwards. It then suffers a small disturbance. For what values of the spin is the initial configuration stable?
If the spin is such that the initial configuration is unstable, what is the lowest angle reached by the symmetry axis in the nutation of the top? Find the maximum and minimum values of the precessional angular velocity .
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A1.3
Part II, 2002 comment(i) Let be the real part of . Establish the following properties of for : (a) ; (b) for ; (c) , uniformly on , as increases to 1 .
(ii) Suppose that , where is the unit circle . By definition, . Let
Show that is a continuous function on , and that .
[You may assume without proof that .]
Show that , uniformly on as increases to 1 , if and only if is a continuous function on .
Show that as increases to 1 .
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A2.3 B2.2
Part II, 2002 comment(i) State and prove the parallelogram law for Hilbert spaces.
Suppose that is a closed linear subspace of a Hilbert space and that . Show that is orthogonal to if and only if 0 is the nearest point to in .
(ii) Suppose that is a Hilbert space and that is a continuous linear functional on with . Show that there is a sequence of unit vectors in with real and .
Show that converges to a unit vector , and that .
Show that is orthogonal to , the null space of , and also that .
Show that , for all .
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A3.3 B3.2
Part II, 2002 comment(i) Suppose that is a decreasing sequence of continuous real-valued functions on a compact metric space which converges pointwise to 0 . By considering sets of the form , for , or otherwise, show that converges uniformly to 0 .
Can the condition that is decreasing be dropped? Can the condition that is compact be dropped? Justify your answers.
(ii) Suppose that is a positive integer. Define polynomials recursively by
Show that , for , and show that converges to uniformly on .
[You may wish to use the identity .]
Suppose that is a closed subalgebra of the algebra of continuous real-valued functions on a compact metric space , equipped with the uniform norm, and suppose that has the property that for each there exists with . Show that there exists such that for all .
Show that for each positive integer , and show that contains the constant functions.
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A4.3
Part II, 2002 commentDefine the distribution function of a non-negative measurable function on the interval . Show that is a decreasing non-negative function on which is continuous on the right.
Define the Lebesgue integral . Show that if and only if almost everywhere.
Suppose that is a non-negative Riemann integrable function on . Show that there are an increasing sequence and a decreasing sequence of non-negative step functions with such that .
Show that the functions and are equal almost everywhere, that is measurable and that the Lebesgue integral is equal to the Riemann integral .
Suppose that is a Riemann integrable function on and that for all . Show that .
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A1.4
Part II, 2002 comment(i) What is a Sylow subgroup? State Sylow's Theorems.
Show that any group of order 33 is cyclic.
(ii) Prove the existence part of Sylow's Theorems.
[You may use without proof any arithmetic results about binomial coefficients which you need.]
Show that a group of order , where and are distinct primes, is not simple. Is it always abelian? Give a proof or a counterexample.
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B1.3
Part II, 2002 commentState Sylow's Theorems. Prove the existence part of Sylow's Theorems.
Show that any group of order 33 is cyclic.
Show that a group of order , where and are distinct primes, is not simple. Is it always abelian? Give a proof or a counterexample.
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A2.4 B2.3
Part II, 2002 comment(i) Show that the ring is Euclidean.
(ii) What are the units in ? What are the primes in ? Justify your answers. Factorize into primes in .
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A3.4
Part II, 2002 comment(i) What does it mean for a ring to be Noetherian? State Hilbert's Basis Theorem. Give an example of a Noetherian ring which is not a principal ideal domain.
(ii) Prove Hilbert's Basis Theorem.
Is it true that if the ring is Noetherian, then so is
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A4.4
Part II, 2002 commentLet be a finite field. Show that there is a unique prime for which contains the field of elements. Prove that contains elements, for some . Show that for all , and hence find a polynomial such that is the splitting field of . Show that, up to isomorphism, is the unique field of size .
[Standard results about splitting fields may be assumed.]
Prove that the mapping sending to is an automorphism of . Deduce that the Galois group Gal is cyclic of order . For which is a subfield of ?
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A B
Part II, 2002 comment(i) Show that, in a region where there is no magnetic field and the charge density vanishes, the electric field can be expressed either as minus the gradient of a scalar potential or as the curl of a vector potential A. Verify that the electric field derived from
is that of an electrostatic dipole with dipole moment .
[You may assume the following identities:
(ii) An infinite conducting cylinder of radius is held at zero potential in the presence of a line charge parallel to the axis of the cylinder at distance , with charge density per unit length. Show that the electric field outside the cylinder is equivalent to that produced by replacing the cylinder with suitably chosen image charges.
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A2.5
Part II, 2002 comment(i) Show that the Lorentz force corresponds to a curvature force and the gradient of a magnetic pressure, and that it can be written as the divergence of a second rank tensor, the Maxwell stress tensor.
Consider the potential field given by , where
referred to cartesian coordinates . Obtain the Maxwell stress tensor and verify that its divergence vanishes.
(ii) The magnetic field in a stellar atmosphere is maintained by steady currents and the Lorentz force vanishes. Show that there is a scalar field such that and . Show further that if is constant, then . Obtain a solution in the form ; describe the structure of this field and sketch its variation in the -direction.
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A3.5 B3.3
Part II, 2002 comment(i) A plane electromagnetic wave in a vacuum has an electric field
referred to cartesian axes . Show that this wave is plane polarized and find the orientation of the plane of polarization. Obtain the corresponding plane polarized magnetic field and calculate the rate at which energy is transported by the wave.
(ii) Suppose instead that
with a constant, . Show that, if the axes are now rotated through an angle so as to obtain an elliptically polarized wave with an electric field
then
Show also that if there is an elliptically polarized wave with
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A4.5
Part II, 2002 commentState the four integral relationships between the electric field and the magnetic field and explain their physical significance. Derive Maxwell's equations from these relationships and show that and can be described by a scalar potential and a vector potential A which satisfy the inhomogeneous wave equations
If the current satisfies Ohm's law and the charge density , show that plane waves of the form
where is a unit vector in the -direction of cartesian axes , are damped. Find an approximate expression for when , where is the electrical conductivity.
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A1.6
Part II, 2002 comment(i) A system in obeys the equations:
where is a positive constant.
By considering the quantity , where and are appropriately chosen, show that if then there is a unique fixed point and a unique limit cycle. How many fixed points are there when ?
(ii) Consider the second order system
where are constants.
(a) Find the fixed points and determine their stability.
(b) Show that if the fixed point at the origin is unstable and then there are no limit cycles.
[You may find it helpful to use the Liénard coordinate .]
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A2.6 B2.4
Part II, 2002 comment(i) Define the terms stable manifold and unstable manifold of a hyperbolic fixed point of a dynamical system. State carefully the stable manifold theorem.
Give an approximation, correct to fourth order in , for the stable and unstable manifolds of the origin for the system
(ii) State, without proof, the centre manifold theorem. Show that the fixed point at the origin of the system
where is a constant, is non-hyperbolic at .
Using new coordinates , find the centre manifold in the form
for constants to be determined. Hence find the evolution equation on the centre manifold in the form
Ignoring higher order terms, give conditions on that guarantee that the origin is asymptotically stable.
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A3.6 B3.4
Part II, 2002 comment(i) Define the Floquet multiplier and Liapunov exponent for a periodic orbit of a dynamical system in . Show that one multiplier is always unity, and that the other is given by
where is the period of the orbit.
The Van der Pol oscillator has a limit cycle . Show using that this orbit is stable.
(ii) Show, by considering the normal form for a Hopf bifurcation from a fixed point of a dynamical system , that in some neighbourhood of the bifurcation the periodic orbit is stable when it exists in the range of for which is unstable, and unstable in the opposite case.
Now consider the system
Show that the fixed point has a Hopf bifurcation when , and is unstable (stable) when .
Suppose that a periodic orbit exists in . Show without solving for the orbit that the result of part (i) shows that such an orbit is unstable. Define a similar result for .
What do you conclude about the existence of periodic orbits when ? Check your answer by applying Dulac's criterion to the system, using the weighting .
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A4.6
Part II, 2002 commentDefine the terms homoclinic orbit, heteroclinic orbit and heteroclinic loop. In the case of a dynamical system that possesses a homoclinic orbit, explain, without detailed calculation, how to calculate its stability.
A second order dynamical system depends on two parameters and . When there is a heteroclinic loop between the points as in the diagram.

When are small there are trajectories that pass close to the fixed points :

By adapting the method used above for trajectories near homoclinic orbits, show that the distances to the stable manifold at on successive returns are related to , , the corresponding distances near , by coupled equations of the form
where any arbitrary constants have been removed by rescaling, and depend on conditions near . Show from these equations that there is a stable heteroclinic orbit if . Show also that in the marginal situation there can be a stable fixed point for small positive if . Explain carefully the form of the orbit of the original dynamical system represented by the solution of the above map when .
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A B1.12
Part II, 2002 comment(i) State the Knaster-Tarski fixed point theorem. Use it to prove the Cantor-Bernstein Theorem; that is, if there exist injections and for two sets and then there exists a bijection .
(ii) Let be an arbitrary set and suppose given a subset of . We define a subset to be -closed just if whenever and then . Show that the set of all -closed subsets of is a complete poset in the inclusion ordering.
Now assume that is itself equipped with a partial ordering .
(a) Suppose satisfies the condition that if then .
Show that if is -closed then implies .
(b) Suppose that satisfies the following condition. Whenever and then there exists such that , and for every we have (i) , and (ii) for some . Let and be -closed subsets of . Show that the set
is -closed.
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B2.11
Part II, 2002 commentExplain what is meant by a structure for a first-order language and by a model for a first-order theory. If is a first-order theory whose axioms are all universal sentences (that is, sentences of the form where is quantifier-free), show that every substructure of a -model is a -model.
Now let be an arbitrary first-order theory in a language , and let be an -structure satisfying all the universal sentences which are derivable from the axioms of . If is a quantifier-free formula (with free variables say) whose interpretation is a nonempty subset of , show that is consistent.
Let be the language obtained from by adjoining a new constant for each element of , and let
Show that has a model. [You may use the Completeness and Compactness Theorems.] Explain briefly why any such model contains a substructure isomorphic to .
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A3.8 B3.11
Part II, 2002 comment(i) Explain briefly what is meant by the terms register machine and computable function.
Let be the universal computable function and a total computable function with . Here and are the unary and binary functions computed by the -th register machine program . Suppose is a total computable function. By considering the function
show that there is a number such that .
(ii) Let be the set of all partial functions . Consider the mapping defined by
(a) Show that any fixed point of is a total function . Deduce that has a unique fixed point.
[The Bourbaki- Witt Theorem may be assumed if stated precisely.]
(b) It follows from standard closure properties of the computable functions that there is a computable function such that
Assuming this, show that there is a total computable function such that
Deduce that the fixed point of is computable.
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A4.8
Part II, 2002 commentLet be a set of primitive propositions. Let denote the set of all compound propositions over , and let be a subset of . Consider the relation on defined by
Prove that is reflexive and transitive. Deduce that if we define by if and only if and , then is an equivalence relation and the quotient is partially ordered by the relation induced by (that is, if and only if , where square brackets denote equivalence classes).
Assuming the result that is a Boolean algebra with lattice operations induced by the logical operations on (that is, , etc.), show that there is a bijection between the following two sets:
(a) The set of lattice homomorphisms .
(b) The set of models of the propositional theory .
Deduce that the completeness theorem for propositional logic is equivalent to the assertion that, for any Boolean algebra with more than one element, there exists a homomorphism .
[You may assume the result that the completeness theorem implies the compactness theorem.]
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B4.10
Part II, 2002 commentExplain what is meant by a well-ordering of a set.
Without assuming Zorn's Lemma, show that the power-set of any well-ordered set can be given a total (linear) ordering.
By a selection function for a set , we mean a function such that for all for all , and if has more than one element. Suppose given a selection function . Given a mapping for some ordinal , we define a subset recursively as follows:
Show that, for any and any ordinal , there exists a function with domain such that .
[It may help to observe that is uniquely determined by and , though you need not show this explicitly.]
Show also that there exists such that, for every with domain is either empty or a singleton.
Deduce that the assertion 'Every set has a selection function' implies that every set can be totally ordered.
[Hartogs' Lemma may be assumed, provided you state it precisely.]
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A1.12 B1.15
Part II, 2002 comment(i) Explain in detail the minimax and Bayes principles of decision theory.
Show that if is a Bayes decision rule for a prior density and has constant risk function, then is minimax.
(ii) Let be independent random variables, with .
Consider estimating by , with loss function
What is the risk function of
Consider the class of estimators of of the form
indexed by . Find the risk function of in terms of , which you should not attempt to evaluate, and deduce that is inadmissible. What is optimal value of ?
[You may assume Stein's Lemma, that for suitably behaved real-valued functions ,
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A2.11 B2.16
Part II, 2002 comment(i) Let be a random variable with density function . Consider testing the simple null hypothesis against the simple alternative hypothesis .
What is the form of the optimal size classical hypothesis test?
Compare the form of the test with the Bayesian test based on the Bayes factor, and with the Bayes decision rule under the 0-1 loss function, under which a loss of 1 is incurred for an incorrect decision and a loss of 0 is incurred for a correct decision.
(ii) What does it mean to say that a family of densities with real scalar parameter is of monotone likelihood ratio?
Suppose has a distribution from a family which is of monotone likelihood ratio with respect to a statistic and that it is required to test against .
State, without proof, a theorem which establishes the existence of a uniformly most powerful test and describe in detail the form of the test.
Let be independent, identically distributed . Find a uniformly most powerful size test of against , and find its power function. Show that we may construct a different, randomised, size test with the same power function for .
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A3.12 B3.15
Part II, 2002 comment(i) Describe in detail how to perform the Wald, score and likelihood ratio tests of a simple null hypothesis given a random sample from a regular oneparameter density . In each case you should specify the asymptotic null distribution of the test statistic.
(ii) Let be an independent, identically distributed sample from a distribution , and let be an estimator of a parameter of .
Explain what is meant by: (a) the empirical distribution function of the sample; (b) the bootstrap estimator of the bias of , based on the empirical distribution function. Explain how a bootstrap estimator of the distribution function of may be used to construct an approximate confidence interval for .
Suppose the parameter of interest is , where is the mean of , and the estimator is , where is the sample mean.
Derive an explicit expression for the bootstrap estimator of the bias of and show that it is biased as an estimator of the true bias of .
Let be the value of the estimator computed from the sample of size obtained by deleting and let . The jackknife estimator of the bias of is
Derive the jackknife estimator for the case , and show that, as an estimator of the true bias of , it is unbiased.
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A4.13 B4.15
Part II, 2002 comment(a) Let be independent, identically distributed random variables from a one-parameter distribution with density function
Explain in detail how you would test
What is the general form of a conjugate prior density for in a Bayesian analysis of this distribution?
(b) Let be independent Poisson random variables, with means and respectively, with known.
Explain why the Conditionality Principle leads to inference about being drawn from the conditional distribution of , given . What is this conditional distribution?
(c) Suppose have distributions as in (b), but that is now unknown.
Explain in detail how you would test against , and describe the optimality properties of your test.
[Any general results you use should be stated clearly, but need not be proved.]
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A1.11 B1.16
Part II, 2002 comment(i) The prices, , of a stock in a binomial model at times are represented by the following binomial tree.

The fixed interest rate per period is and the probability that the stock price increases in a period is . Find the price at time 0 of a European call option with strike price 78 and expiry time
Explain briefly the ideas underlying your calculations.
(ii) Consider an investor in a one-period model who may invest in assets, all of which are risky, with a random return vector having mean and positivedefinite covariance matrix ; assume that not all the assets have the same expected return. Show that any minimum-variance portfolio is equivalent to the investor dividing his wealth between two portfolios, the global minimum-variance portfolio and the diversified portfolio, both of which should be specified clearly in terms of and .
Now suppose that where are independent random variables with having the exponential distribution with probability density function , where . Determine the global minimum-variance portfolio and the diversified portfolio explicitly.
Consider further the situation when the investor has the utility function , where denotes his wealth. Suppose that he acts to maximize the expected utility of his final wealth, and that his initial wealth is . Show that he now divides his wealth between the diversified portfolio and the uniform portfolio, in which wealth is apportioned equally between the assets, and determine the amounts that he invests in each.
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A3.11 B3.16
Part II, 2002 comment(i) Explain briefly what it means to say that a stochastic process is a standard Brownian motion.
Let be a standard Brownian motion and let be real numbers. What condition must and satisfy to ensure that the process is a martingale? Justify your answer carefully.
(ii) At the beginning of each of the years an investor has income , of which he invests a proportion , and consumes the rest during the year. His income at the beginning of the next year is
where are independent positive random variables with finite means and is a constant. He decides on after he has observed both and at the beginning of year , but at that time he does not have any knowledge of the value of , for any . The investor retires in year and consumes his entire income during that year. He wishes to determine the investment policy that maximizes his expected total consumption
Prove that the optimal policy may be expressed in terms of the numbers , where , for , and determine the optimal expected total consumption.
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A4.12 B4.16
Part II, 2002 commentWrite an essay on the Black-Scholes formula for the price of a European call option on a stock. Your account should include a derivation of the formula and a careful analysis of its dependence on the parameters of the model.
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A2.13 B2.21
Part II, 2002 comment(i) A Hamiltonian has energy eigenvalues and corresponding non-degenerate eigenstates . Show that under a small change in the Hamiltonian ,
and derive the related formula for the change in the energy eigenvalue to first and second order in .
(ii) The Hamiltonian for a particle moving in one dimension is , where and is small. Show that
and hence that
to second order in .
Deduce that is independent of the particular state and explain why this change in energy is exact to all orders in .
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A3.13 B3.21
Part II, 2002 comment(i) Two particles with angular momenta and basis states are combined to give total angular momentum and basis states . State the possible values of and show how a state with can be constructed. Briefly describe, for a general allowed value of , what the Clebsch-Gordan coefficients are.
(ii) If the angular momenta and are both 1 show that the combined state is

Determine the corresponding expressions for the combined states and , assuming that they are respectively antisymmetric and symmetric under interchange of the two particles.
If the combined system is in state what is the probability that measurements of the -component of angular momentum for either constituent particle will give the value of 1 ?
Hint:
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A4.15 B4.22
Part II, 2002 commentDiscuss the consequences of indistinguishability for a quantum mechanical state consisting of two identical, non-interacting particles when the particles have (a) spin zero, (b) spin 1/2.
The stationary Schrödinger equation for one particle in the potential
has normalized, spherically symmetric, real wave functions and energy eigenvalues with . What are the consequences of the Pauli exclusion principle for the ground state of the helium atom? Assuming that wavefunctions which are not spherically symmetric can be ignored, what are the states of the first excited energy level of the helium atom?
[You may assume here that the electrons are non-interacting.]
Show that, taking into account the interaction between the two electrons, the estimate for the energy of the ground state of the helium atom is
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A1.15 B1.24
Part II, 2002 comment(i) Given a covariant vector field , define the curvature tensor by
Express in terms of the Christoffel symbols and their derivatives. Show that
Further, by setting , deduce that
(ii) Write down an expression similar to (*) given in Part (i) for the quantity
and hence show that
Define the Ricci tensor, show that it is symmetric and write down the contracted Bianchi identities.
In certain spacetimes of dimension takes the form
Obtain the Ricci tensor and Ricci scalar. Deduce that is a constant in such spacetimes if the dimension is greater than 2 .
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A2.15 B2.23
Part II, 2002 comment(i) Consider the line element describing the interior of a star,
defined for by
and
Here is the mass of the star, and is defined to be .
The star is made of a perfect fluid with energy-momentum tensor
Here is the 4 -velocity of the fluid which is at rest, the density is constant throughout the star and the pressure depends only on the radial coordinate. Write down the Einstein field equations and show that (in geometrical units with ) they may equivalently be written as
(ii) Using the formulae below, or otherwise, show that for one has
[The non-zero components of the Ricci tensor are:
Note that
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A4.17 B4.25
Part II, 2002 commentWith respect to the Schwarzschild coordinates , the Schwarzschild geometry is given by
where is the Schwarzschild radius and is the Schwarzschild mass. Show that, by a suitable choice of , the general geodesic can regarded as moving in the equatorial plane . Obtain the equations governing timelike and null geodesics in terms of , where .
Discuss light bending and perihelion precession in the solar system.
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A1.20 B1.20
Part II, 2002 comment(i) Let be an symmetric real matrix with distinct eigenvalues and corresponding eigenvectors , where . Given , the sequence is generated in the following manner. We set
Show that if
where is a real scalar and is chosen so that , then
Give an explicit expression for .
(ii) Use the above result to prove that, if is small,
and obtain the numbers and .
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A2.19 B2.19
Part II, 2002 comment(i)
Given the finite-difference method
define
Prove that this method is stable if and only if
[You may quote without proof known properties of the Fourier transform.]
(ii) Find the range of the parameter such that the method
is stable. Supposing that this method is used to solve the diffusion equation for , determine the order of magnitude of the local error as a power of .
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A3.19 B3.20
Part II, 2002 comment(i) Determine the order of the multistep method
for the solution of ordinary differential equations for different choices of in the range .
(ii) Prove that no such choice of results in a method whose linear stability domain includes the interval .
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A4.22 B4.20
Part II, 2002 commentWrite an essay on the method of conjugate gradients. You should describe the algorithm, present an analysis of its properties and discuss its advantages.
[Any theorems quoted should be stated precisely but need not be proved.]
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B1.5
Part II, 2002 commentProve that every graph on vertices with minimal degree is Hamiltonian. For each , give an example to show that this result does not remain true if we weaken the condition to ( even) or ( odd).
Now let be a connected graph (with at least 2 vertices) without a cutvertex. Does Hamiltonian imply Eulerian? Does Eulerian imply Hamiltonian? Justify your answers.
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B2.5
Part II, 2002 commentState and prove the local inequality. Explain carefully when equality holds.
Define the colex order and state the Kruskal-Katona theorem. Deduce that, if and are fixed positive integers with , then for every we have
By a suitable choice of and , show that this result does not remain true if we replace the lower shadow with the upper shadow .
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B4.1
Part II, 2002 commentWrite an essay on Ramsey theory. You should include the finite and infinite versions of Ramsey's theorem, together with a discussion of upper and lower bounds in the finite case.
[You may restrict your attention to colourings by just 2 colours.]
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B1.6
Part II, 2002 commentConstruct the character table of the symmetric group , explaining the steps in your construction.
Use the character table to show that the alternating group is the only non-trivial normal subgroup of .
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B2.6
Part II, 2002 commentState and prove Schur's Lemma. Deduce that the centre of a finite group with a faithful irreducible complex representation is cyclic and that consists of scalar transformations.
Let be the subgroup of order 18 of the symmetric group given by
Show that has a normal subgroup of order 9 and four normal subgroups of order 3 . By considering quotients, show that has two representations of dimension 1 and four inequivalent irreducible representations of degree 2 . Deduce that has no faithful irreducible complex representations.
Show finally that if is a finite group with trivial centre and is a subgroup of with non-trivial centre, then any faithful representation of is reducible on restriction to .
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B3.5
Part II, 2002 commentLet be a finite group acting on a finite set . Define the permutation representation of and compute its character . Prove that equals the number of orbits of on . If acts also on the finite set , with character , show that equals the number of orbits of on .
Now let be the symmetric group acting naturally on the set , and let be the set of all -element subsets of . Let be the permutation character of on . Prove that
Deduce that the class functions
are irreducible characters of , for .
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B4.2
Part II, 2002 commentWrite an essay on the representation theory of .
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B1.7
Part II, 2002 commentLet be a finite extension of fields and let be the group of -automorphisms of . State a result relating the order of to the degree .
Now let be the field of rational functions in four variables over a field and let where are the elementary symmetric polynomials in . Show that the degree ! and deduce that is the fixed field of the natural action of the symmetric group on .
Show that has a cubic minimum polynomial over . Let be the dihedral group generated by the permutations and . Show that the fixed field of is . Find the fixed field of the subgroup .
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B3.6
Part II, 2002 commentShow that the polynomial has no rational roots. Show that the splitting field of over the finite field is an extension of degree 4 . Hence deduce that is irreducible over the rationals. Prove that has precisely two (non-multiple) roots over the finite field . Find the Galois group of over the rationals.
[You may assume any general results you need including the fact that is the only index 2 subgroup of .]
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B4.3
Part II, 2002 commentSuppose are fields and are distinct embeddings of into . Prove that there do not exist elements of (not all zero) such that for all . Deduce that if is a finite extension of fields, and are distinct -automorphisms of , then .
Suppose now that is a Galois extension of with Galois group cyclic of order , where is not divisible by the characteristic. If contains a primitive th root of unity, prove that is a radical extension of . Explain briefly the relevance of this result to the problem of solubility of cubics by radicals.
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B1.8
Part II, 2002 commentWhat is meant by a "bump function" on ? If is an open subset of a manifold , prove that there is a bump function on with support contained in .
Prove the following.
(i) Given an open covering of a compact manifold , there is a partition of unity on subordinate to .
(ii) Every compact manifold may be embedded in some Euclidean space.
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B2.7
Part II, 2002 commentState, giving your reasons, whether the following are true or false.
(a) Diffeomorphic connected manifolds must have the same dimension.
(b) Every non-zero vector bundle has a nowhere-zero section.
(c) Every projective space admits a volume form.
(d) If a manifold has Euler characteristic zero, then is orientable.
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B4.4
Part II, 2002 commentState and prove Stokes' Theorem for compact oriented manifolds-with-boundary.
[You may assume results relating local forms on the manifold with those on its boundary provided you state them clearly.]
Deduce that every differentiable map of the unit ball in to itself has a fixed point.
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B2.8
Part II, 2002 commentShow that the fundamental group of the Klein bottle is infinite. Show that contains an abelian subgroup of finite index. Show that is not abelian.
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B3.7
Part II, 2002 commentFor a finite simplicial complex , let denote the rank of the finitely generated abelian group . Define the Euler characteristic by the formula
Let denote the number of -simplices in , for each . Show that
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B2.9
Part II, 2002 commentLet . By Dedekind's theorem, or otherwise, show that the ideal equations
hold in , where . Deduce that has class number 2 .
Verify that is the fundamental unit in . Hence show that the complete solution in integers of the equation is given by
Calculate the particular solution for .
[It can be assumed that the Minkowski constant for is .]
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B4.5
Part II, 2002 commentState the Mayer-Vietoris theorem for a finite simplicial complex which is the union of closed subcomplexes and . Define all the maps in the long exact sequence. Prove that the sequence is exact at the term , for every .
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B1.9
Part II, 2002 commentExplain what is meant by an integral basis of a number field . Give an expression for the discriminant of in terms of the traces of the .
Let . By computing the traces , where runs through the three quadratic subfields of , show that the algebraic integers in have the form , where and are Gaussian integers. By further computing the norm , where , show that and are even and that . Hence prove that an integral basis for is .
Calculate the discriminant of .
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B4.6
Part II, 2002 commentWrite an essay on one the following topics.
(i) Dirichlet's unit theorem and the Pell equation.
(ii) Ideals and the fundamental theorem of arithmetic.
(iii) Dedekind's theorem and the factorisation of primes. (You should treat explicitly either the case of quadratic fields or that of the cyclotomic field.)
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B1.10
Part II, 2002 commentLet be a Hilbert space and let . Define what it means for to be bounded below. Prove that, if for some , then is bounded below.
Prove that an operator is invertible if and only if both and are bounded below.
Let be the sequence space . Define the operators on by setting
for all . Check that but . Let . For each , explain why is invertible, and define
Show that, for all , we have , but . Deduce that, for all , the operator is bounded below, but is not invertible. Deduce also that .
Let with , and for , define the element of by
Prove that but that as . Deduce that, for is not bounded below.
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B3.8
Part II, 2002 commentLet be an infinite-dimensional, separable Hilbert space. Let be a compact linear operator on , and let be a non-zero, approximate eigenvalue of . Prove that is an eigenvalue, and that the corresponding eigenspace is finite-dimensional.
Let be a compact, self-adjoint operator on . Prove that there is an orthonormal basis of , and a sequence in , such that (i) and (ii) as .
Now let be compact, self-adjoint and injective. Let be a bounded self-adjoint operator on such that . Prove that has an orthonormal basis , where, for every is an eigenvector, both of and of .
[You may assume, without proof, results about self-adjoint operators on finite-dimensional spaces.]
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B4.7
Part II, 2002 commentThroughout this question, is an infinite-dimensional, separable Hilbert space. You may use, without proof, any theorems about compact operators that you require.
Define a Fredholm operator , on a Hilbert space , and define the index of .
(i) Prove that if is Fredholm then is closed.
(ii) Let and let have finite rank. Prove that also has finite rank.
(iii) Let , where is the identity operator on and has finite rank; let . By considering and (or otherwise) prove that is Fredholm with ind .
(iv) Let be Fredholm with ind . Prove that , where is invertible and has finite rank.
[You may wish to note that effects an isomorphism from onto ; also ker and have the same finite dimension.]
(v) Deduce from (iii) and (iv) that is Fredholm with ind if and only if with invertible and compact.
(vi) Explain briefly, by considering suitable shift operators on (i.e. not using any theorems about Fredholm operators) that, for each , there is a Fredholm operator on with ind .
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B1.11
Part II, 2002 comment(a) Define the notions of (abstract) Riemann surface, holomorphic map, and biholomorphic map between Riemann surfaces.
(b) Prove the following theorem on the local form of a holomorphic map.
For a holomorphic map between Riemann surfaces, which is not constant near a point , there exist neighbourhoods of in and of in , together with biholomorphic identifications , such that , for all .
(c) Prove further that a non-constant holomorphic map between compact, connected Riemann surfaces is surjective.
(d) Deduce from (c) the fundamental theorem of algebra.
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B3.9
Part II, 2002 commentLet be two non-zero complex numbers with . Let be the lattice . A meromorphic function on is elliptic if , for all and . The Weierstrass functions are defined by the following properties:
-
is elliptic, has double poles at the points of and no other poles, and near 0
-
, and near 0 ;
-
is odd, and , and as .
Prove the following
(a) , and hence and , are uniquely determined by these properties. You are not expected to prove the existence of , and you may use Liouville's theorem without proof.
(b) , and , for some constants .
(c) is holomorphic, has simple zeroes at the points of , and has no other zeroes.
(d) Given and in with , the function
is elliptic.
(e) .
(f) Deduce from (e), or otherwise, that .
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B4.8
Part II, 2002 commentA holomorphic map between Riemann surfaces is called a covering map if every has a neighbourhood for which breaks up as a disjoint union of open subsets on which is biholomorphic.
(a) Suppose that is any holomorphic map of connected Riemann surfaces, is simply connected and is a covering map. By considering the lifts of paths from to , or otherwise, prove that lifts to a holomorphic map , i.e. that there exists an with .
(b) Write down a biholomorphic map from the unit disk onto a half-plane. Show that the unit disk uniformizes the punctured unit disk by constructing an explicit covering map .
(c) Using the uniformization theorem, or otherwise, prove that any holomorphic map from to a compact Riemann surface of genus greater than one is constant.
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B2.10
Part II, 2002 commentFor , let be the (irreducible) projective plane curve over an algebraically closed field of characteristic zero.
Show that is smooth (non-singular). For , let be the morphism . Determine the degree of , its points of ramification and the corresponding ramification indices.
Applying the Riemann-Hurwitz formula to , determine the genus of .
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B3.10
Part II, 2002 commentLet be an irreducible polynomial of degree (over an algebraically closed field of characteristic zero) and the corresponding affine plane curve. Assume that is smooth (non-singular) and that the projectivization of intersects the line at infinity in distinct points. Show that is smooth and determine the divisor of the rational differential on . Deduce a formula for the genus of .
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B4.9
Part II, 2002 commentWrite an essay on the Riemann-Roch theorem and some of its applications.
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B1.13
Part II, 2002 commentState and prove Dynkin's -system lemma.
Let be a probability space and let be a sequence of independent events such that . Let . Prove that
for all .
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B2.12
Part II, 2002 commentLet be a sequence of non-negative random variables on a common probability space with , such that almost surely. Determine which of the following statements are necessarily true, justifying your answers carefully: (a) as ; (b) as ; (c) as ; (d) as .
[Standard limit theorems for integrals, and results about uniform integrability, may be used without proof provided that they are clearly stated.]
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B3.12
Part II, 2002 commentDerive the characteristic function of a real-valued random variable which is normally distributed with mean and variance . What does it mean to say that an -valued random variable has a multivariate Gaussian distribution? Prove that the distribution of such a random variable is determined by its -valued) mean and its covariance matrix.
Let and be random variables defined on the same probability space such that has a Gaussian distribution. Show that and are independent if and only if . Show that, even if they are not independent, one may always write for some constant and some random variable independent of .
[The inversion theorem for characteristic functions and standard results about independence may be assumed.]
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B4.11
Part II, 2002 commentState Birkhoff's Almost Everywhere Ergodic Theorem for measure-preserving transformations. Define what it means for a sequence of random variables to be stationary. Explain briefly how the stationarity of a sequence of random variables implies that a particular transformation is measure-preserving.
A bag contains one white ball and one black ball. At each stage of a process one ball is picked from the bag (uniformly at random) and then returned to the bag together with another ball of the same colour. Let be a random variable which takes the value 0 if the th ball added to the bag is white and 1 if it is black.
(a) Show that the sequence is stationary and hence that the proportion of black balls in the bag converges almost surely to some random variable .
(b) Find the distribution of .
[The fact that almost-sure convergence implies convergence in distribution may be used without proof.]
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B2.13
Part II, 2002 commentTwo enthusiastic probability students, Ros and Guil, sit an examination which starts at time 0 and ends at time ; they both decide to use the time to attempt a proof of a difficult theorem which carries a lot of extra marks.
Ros' strategy is to write the proof continuously at a constant speed lines per unit time. In a time interval of length he has a probability of realising he has made a mistake. If that happens he instantly panics, erases everything he has written and starts all over again.
Guil, on the other hand, keeps cool and thinks carefully about what he is doing. In a time interval of length , he has a probability of writing the next line of proof and for each line he has written a probability of finding a mistake in that line, independently of all other lines he has written. When a mistake is found, he erases that line and carries on as usual, hoping for the best.
Both Ros and Guil realise that, even if they manage to finish the proof, they will not recognise that they have done so and will carry on writing as much as they can.
(a) Calculate , the probability that, for Ros, the length of his completed proof at time is at least .
(b) Let be the probability that Guil has lines of proof at time . Show that
where .
(c) Suppose now that every time Ros starts all over again, the time until the next mistake has distribution , independently of the past history. Write down a renewal-type integral equation satisfied by , the expected length of Ros' proof at time . What is the expected length of proof produced by him at the end of the examination if is the exponential distribution with mean ?
(d) What is the expected length of proof produced by Guil at the end of the examination if each line that he writes survives for a length of time with distribution , independently of all other lines?
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B3.13
Part II, 2002 comment(a) Define a renewal process and a discrete renewal process.
(b) State and prove the Discrete Renewal Theorem.
(c) The sequence satisfies
for some collection of non-negative numbers summing to 1 . Let . Show that
Give a probabilistic interpretation of the numbers and .
(d) Let the sequence be given by
How is this related to the simple symmetric random walk on the integers starting from the origin, and its subsequent returns to the origin? Determine in this case, either by calculating or by showing that satisfies the quadratic equation
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B4.12
Part II, 2002 commentDefine a Poisson random measure. State and prove the Product Theorem for the jump times of a Poisson process with constant rate and independent random variables with law . Write down the corresponding result for a Poisson process in a space with rate when we associate with each an independent random variable with density .
Prove Campbell's Theorem, i.e. show that if is a Poisson random measure on the space with intensity measure and is a bounded measurable function then
where
Stars are scattered over three-dimensional space in a Poisson process with density . Masses of the stars are independent random variables; the mass of a star at has the density . The gravitational potential at the origin is given by
where is a constant. Find the moment generating function .
A galaxy occupies a sphere of radius centred at the origin. The density of stars is for points inside the sphere; the mass of each star has the exponential distribution with mean . Calculate the expected potential due to the galaxy at the origin. Let be a positive constant. Find the distribution of the distance from the origin to the nearest star whose contribution to the potential is at least .
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B1.14
Part II, 2002 comment(a) Define the entropy and the mutual entropy of random variables and . Prove the inequality
[You may assume the Gibbs inequality.]
(b) Let be a random variable and let be a random vector.
(i) Prove or disprove by producing a counterexample the inequality
first under the assumption that are independent random variables, and then under the assumption that are conditionally independent given .
(ii) Prove or disprove by producing a counterexample the inequality
first under the assumption that are independent random variables, and then under the assumption that are conditionally independent given .
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B2.14
Part II, 2002 commentDefine the binary Hamming code of length and its dual. Prove that the Hamming code is perfect. Prove that in the dual code:
(i) The weight of any non-zero codeword equals ;
(ii) The distance between any pair of words equals .
[You may quote results from the course provided that they are carefully stated.]
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B4.13
Part II, 2002 commentDefine the Huffman binary encoding procedure and prove its optimality among decipherable codes.
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B2.15
Part II, 2002 commentState Pontryagin's maximum principle (PMP) for the problem of minimizing
where ; here, and are given, and is unconstrained.
Consider the two-dimensional problem in which , and . Show that, by use of a variable , one can rewrite this problem as an equivalent one-dimensional problem.
Use PMP to solve this one-dimensional problem, showing that the optimal control can be expressed as , where .
Express in a feedback form of for some .
Suppose that the initial state is perturbed by a small amount to . Give an expression (in terms of and ) for the increase in minimal cost.
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B3.14
Part II, 2002 commentConsider a scalar system with , where is a sequence of independent random variables, uniform on the interval , with . We wish to choose to minimize the expected value of
where is chosen knowing but not . Prove that the minimal expected cost can be written and derive a recurrence for calculating .
How does your answer change if is constrained to lie in the set
Consider a stopping problem for which there are two options in state :
(1) stop: paying a terminal cost ; no further costs are incurred;
(2) continue: choosing , paying , and moving to state
Consider the problem of minimizing total expected cost subject to the constraint that no more than continuation steps are allowed. Suppose . Show that an optimal policy stops if and only if either continuation steps have already been taken or .
[Hint: Use induction on to show that a one-step-look-ahead rule is optimal. You should not need to find the optimal for the continuation steps.]
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B4.14
Part II, 2002 commentA discrete-time decision process is defined on a finite set of states as follows. Upon entry to state at time the decision-maker observes a variable . He then chooses the next state freely within , at a cost of . Here is a sequence of integer-valued, identically distributed random variables. Suppose there exist and such that for all
Let denote a policy. Show that
At the start of each month a boat manufacturer receives orders for 1, 2 or 3 boats. These numbers are equally likely and independent from month to month. He can produce boats in a month at a cost of units. All orders are filled at the end of the month in which they are ordered. It is possible to make extra boats, ending the month with a stock of unsold boats, but cannot be more than 2 , and a holding cost of is incurred during any month that starts with unsold boats in stock. Write down an optimality equation that can be used to find the long-run expected average-cost.
Let be the policy of only ever producing sufficient boats to fill the present month's orders. Show that it is optimal if and only if .
Suppose . Starting from , what policy is obtained after applying one step of the policy-improvement algorithm?
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B1.17
Part II, 2002 commentLet be the map of the closed interval to itself given by
Sketch the graphs of and (without proof) of , find their fixed points, and determine which of the fixed points of are attractors. Does your argument work for
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B3.17
Part II, 2002 commentLet be a finite alphabet of letters and either the semi-infinite space or the doubly infinite space of sequences whose elements are drawn from . Define the natural topology on . If is a set of words, denote by the subspace of consisting of those sequences none of whose subsequences is in . Prove that is a closed subspace of ; and state and prove a necessary and sufficient condition for a closed subspace of to have the form for some .
what is the space ?
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B4.17
Part II, 2002 commentLet be a metric space, a map of to itself and a point of . Define an attractor for and an omega point of the orbit of under .
Let be the map of to itself given by
where is so small that for all , and let be the map of to itself induced by . What points if any are
(a) attractors for ,
(b) omega points of the orbit of some point under ?
Is the cycle an attractor?
In the notation of the first two sentences, let be a cycle of order and assume that is continuous. Prove that is an attractor for if and only if each point of is an attractor for .
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B1.18
Part II, 2002 comment(a) Solve the equation, for a function ,
together with the boundary condition on the -axis:
Find for which real numbers it is possible to solve with the following boundary condition specified on the line :
Explain your answer in terms of the notion of characteristic hypersurface, which should be defined.
(b) Solve the equation
with the boundary condition on the -axis
in the domain . Sketch the characteristics.
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B2.17
Part II, 2002 comment(a) Define the convolution of two functions. Write down a formula for a solution to the initial value problem
together with the boundary condition
for a bounded continuous function on . Comment briefly on the uniqueness of the solution.
(b) State and prove the Duhamel principle giving the solution (for ) to the equation
together with the boundary condition
in terms of your answer to (a).
(c) Show that if is the solution to
together with the boundary condition
with for all then for all .
Finally show that if in addition there exists a point at which there is strict inequality in the assumption i.e.
then in fact
whenever .
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B3.18
Part II, 2002 commentDefine the Schwartz space and the space of tempered distributions . State the Fourier inversion theorem for the Fourier transform of a Schwartz function.
Consider the initial value problem:
for in the Schwartz space .
Show that the solution can be written as
where
and
State the Plancherel-Parseval theorem and hence deduce that
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B4.18
Part II, 2002 commentDiscuss the notion of fundamental solution for a linear partial differential equation with constant coefficients.
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B1.19
Part II, 2002 commentState the Riemann-Lebesgue lemma as applied to the integral
where is continuous and .
Use this lemma to show that, as ,
where is holomorphic, and . You should explain each step of your argument, but detailed analysis is not required.
Hence find the leading order asymptotic behaviour as of
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B2.18
Part II, 2002 commentShow that
where is real and positive, and denotes the Cauchy principal value; the principal branches of etc. are implied. Deduce that
and that
Use to show that, if , then
What is the value of this integral if ?
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B3.19
Part II, 2002 commentShow that the equation
has solutions of the form
Give examples of possible choices of to provide two independent solutions, assuming . Distinguish between the cases and . Comment on the case and on the case that is an odd integer.
Show that, if , there is a solution that is bounded as , and that, in this limit,
where is a constant.
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B4.19
Part II, 2002 commentLet
where is real, is real and non-zero, and the path of integration runs up the imaginary axis. Show that, if ,
as and sketch the relevant steepest descent path.
What is the corresponding result if ?
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B1.21
Part II, 2002 commentExplain how one can write Maxwell's equations in relativistic form by introducing an antisymmetric field strength tensor .
In an inertial frame , the electric and magnetic fields are and . Suppose that there is a second inertial frame moving with velocity along the -axis relative to . Derive the rules for finding the electric and magnetic fields and in the frame . Show that and are invariant under Lorentz transformations.
Suppose that and , where . At what velocity must an observer be moving in the frame for the electric and magnetic fields to appear to be parallel?
Comment on the case .
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B2.20
Part II, 2002 commentA particle of rest mass and charge moves in an electromagnetic field given by a potential along a trajectory , where is the proper time along the particle's worldline. The action for such a particle is
Show that the Euler-Lagrange equations resulting from this action reproduce the relativistic equation of motion for the particle.
Suppose that the particle is moving in the electrostatic field of a fixed point charge with radial electric field given by
Show that one can choose a gauge such that and only . Find .
Assume that the particle executes planar motion, which in spherical polar coordinates can be taken to be in the plane . Derive the equations of motion for and .
By using the fact that , find the equation of motion for , and hence show that the shape of the orbit is described by
where and are constants of integration and is to be determined.
By putting or otherwise, show that if then the orbits are bounded and generally not closed, and show that the angle between successive minimal values of is .
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B4.21
Part II, 2002 commentDerive Larmor's formula for the rate at which radiation is produced by a particle of charge moving along a trajectory .
A non-relativistic particle of mass , charge and energy is incident along a radial line in a central potential . The potential is vanishingly small for very large, but increases without bound as . Show that the total amount of energy radiated by the particle is
where .
Suppose that is the Coulomb potential . Evaluate .
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B1.22
Part II, 2002 commentA simple model for a rubber molecule consists of a one-dimensional chain of links each of fixed length and each of which is oriented in either the positive or negative direction. A unique state of the molecule is designated by giving the orientation of each link. If there are links oriented in the positive direction and links oriented in the negative direction then and the length of the molecule is . The length of the molecule associated with state is .
What is the range of ?
What is the number of states with fixed?
Consider an ensemble of copies of the molecule in which members are in state and write down the expression for the mean length .
By introducing a Lagrange multiplier for show that the most probable configuration for the with given length is found by maximizing
Hence show that the most probable configuration is given by
where is the probability for finding an ensemble member in the state and is the partition function which should be defined.
Show that can be expressed as
where the meaning of should be explained.
Hence show that is given by
and therefore that the free energy for the system is
Show that is determined by
and hence that the equation of state is
What are the independent variables on which depends?
Explain why the tension in the rubber molecule is .
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B3.22
Part II, 2002 commentA system consisting of non-interacting bosons has single-particle levels uniquely labelled by with energies . Show that the free energy in the grand canonical ensemble is
What is the maximum value for ?
A system of bosons in a large volume has one energy level of energy zero and a large number of energy levels of the same energy , where takes the form with a positive constant. What are the dimensions of
Show that the free energy is
The numbers of particles with energies are respectively . Write down expressions for in terms of .
At temperature what is the maximum number of bosons in the normal phase (the state with energy )? Explain what happens when .
Given and calculate the transition temperature at which Bose condensation occurs.
For show that . What is the value of for ?
Calculate the mean energy for (a) (b) , and show that the heat capacity of the system at constant volume is
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B4.23
Part II, 2002 commentA perfect gas in equilibrium in a volume has quantum stationary states with energies . In a Boltzmann distribution, the probability that the system is in state is . The entropy is defined to be .
For two nearby states establish the equation
where and should be defined.
For reversible changes show that
where is the amount of heat transferred in the exchange.
Define , the heat capacity at constant volume.
A system with constant heat capacity initially at temperature is heated at constant volume to a temperature . Show that the change in entropy is .
Explain what is meant by isothermal and adiabatic transitions.
Briefly, describe the Carnot cycle and define its efficiency. Explain briefly why no heat engine can be more efficient than one whose operation is based on a Carnot cycle.
Three identical bodies with constant heat capacity at fixed volume , are initially at temperatures , respectively. Heat engines operate between the bodies with no input of work or heat from the outside and the respective temperatures are changed to , the volume of the bodies remaining constant. Show that, if the heat engines operate on a Carnot cycle, then
where and .
Hence show that the maximum temperature to which any one of the bodies can be raised is where
Show that a solution is if initially . Do you expect there to be any other solutions?
Find if initially .
[Hint: Choose to maximize one temperature and impose the constraints above using Lagrange multipliers. ]
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B1.23
Part II, 2002 commentA quantum system, with Hamiltonian , has continuous energy eigenstates for all , and also a discrete eigenstate , with . A time-independent perturbation , such that , is added to . If the system is initially in the state obtain the formula for the decay rate
where is the density of states.
[You may assume that behaves like for large .]
Assume that, for a particle moving in one dimension,
where , and is constant. Obtain in this case.
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B2.22
Part II, 2002 commentDefine the reciprocal lattice for a lattice with lattice vectors .
A beam of electrons, with wave vector , is incident on a Bravais lattice with a large number of atoms, . If the scattering amplitude for scattering on an individual atom in the direction is , show that the scattering amplitude for the whole lattice
Derive the formula for the differential cross section
obtaining an explicit form for . Show that is strongly peaked when , a reciprocal lattice vector. Show that this leads to the Bragg formula , where is the scattering angle, the electron wavelength and the separation between planes of atoms in the lattice.
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B3.23
Part II, 2002 commentA periodic potential is expressed as , where are reciprocal lattice vectors and . In the nearly free electron model explain why it is appropriate, near the boundaries of energy bands, to consider a Bloch wave state
where is a free electron state for wave vector , and the sum is restricted to reciprocal lattice vectors such that . Obtain a determinantal formula for the possible energies corresponding to Bloch wave states of this form.
[You may take and assume for any .]
Suppose the sum is restricted to just and . Show that there is a gap between energy bands. Setting , show that there are two Bloch wave states with energies near the boundaries of the energy bands
What is meant by effective mass? Determine the value of the effective mass at the top and the bottom of the adjacent energy bands if is parallel to .
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B4.24
Part II, 2002 commentExplain the variational method for computing the ground state energy for a quantum Hamiltonian.
For the one-dimensional Hamiltonian
obtain an approximate form for the ground state energy by considering as a trial state the state defined by , where and .
[It is useful to note that .]
Explain why the states may be used as trial states for calculating the first excited energy level.
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B1.25
Part II, 2002 commentState the minimum dissipation theorem for Stokes flow in a bounded domain.
Fluid of density and viscosity fills an infinite cylindrical annulus between a fixed cylinder and a cylinder which rotates about its axis with constant angular velocity . In cylindrical polar coordinates , the fluid velocity is . The Reynolds number is not necessarily small. Show that , where and are constants to be determined.
[You may assume that and ]
Show that the outer cylinder exerts a couple per unit length on the fluid, where
[You may assume that, in standard notation, .]
Suppose now that and that the cylinder is replaced by a fixed cylinder whose cross-section is a square of side centred on , all other conditions being unchanged. The flow may still be assumed steady. Explaining your argument carefully, show that the couple now required to maintain the motion of the outer cylinder is greater than .
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B2.24
Part II, 2002 commentA thin layer of liquid of kinematic viscosity flows under the influence of gravity down a plane inclined at an angle to the horizontal . With origin on the plane, and axes down the line of steepest slope and normal to the plane, the free surface is given by , where . The pressure distribution in the liquid may be assumed to be hydrostatic. Using the approximations of lubrication theory, show that
Now suppose that
where
and and are constants with . Show that, to leading order,
where and are constants to be determined.
Explain in physical terms the meaning of this solution.
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B3.24
Part II, 2002 comment(i) Suppose that, with spherical polar coordinates, the Stokes streamfunction
represents a Stokes flow and thus satisfies the equation , where
Show that the possible values of are and . For which of these values is the corresponding flow irrotational? Sketch the streamlines of the flow for the case .
(ii) A spherical drop of liquid of viscosity , radius and centre at , is suspended in another liquid of viscosity which flows with streamfunction
far from the drop. The two liquids are of equal densities, surface tension is sufficiently strong to keep the drop spherical, and inertia is negligible. Show that
and obtain four equations determining the constants and . (You need not solve these equations.)
[You may assume, with standard notation, that
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B4.26
Part II, 2002 commentWrite an essay on boundary-layer theory and its application to the generation of lift in aerodynamics.
You should include discussion of the derivation of the boundary-layer equation, the similarity transformation leading to the Falkner-Skan equation, the influence of an adverse pressure gradient, and the mechanism(s) by which circulation is generated in flow past bodies with a sharp trailing edge.
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B1.26
Part II, 2002 commentStarting from the equations governing sound waves linearized about a state with density and sound speed , derive the acoustic energy equation, giving expressions for the local energy density and energy flux .
A sphere executes small-amplitude vibrations, with its radius varying according to
with . Find an expression for the velocity potential of the sound, . Show that the time-averaged rate of working by the surface of the sphere is
Calculate the value at of the dimensionless ratio , where the overbars denote time-averaged values, and comment briefly on the limits and .
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B2 25
Part II, 2002 commentStarting from the equations for one-dimensional unsteady flow of a perfect gas of uniform entropy, show that the Riemann invariants,
are constant on characteristics given by , where is the velocity of the gas, is the local speed of sound and is the specific heat ratio.
Such a gas initially occupies the region to the right of a piston in an infinitely long tube. The gas and the piston are initially at rest. At time the piston starts moving to the left at a constant speed . Find and in the three regions
where . What is the largest value of for which is positive throughout region (iii)? What happens if exceeds this value?
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B3.25
Part II, 2002 commentConsider the equation
Find the dispersion relation for waves of frequency and wavenumber . Do the wave crests move faster or slower than a packet of waves?
Write down the solution with initial value
where is real and .
Use the method of stationary phase to obtain an approximation to for large , with having the constant value . Explain, using the notion of group velocity, the constraint that must be placed on .
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B4.27
Part II, 2002 commentWrite down the equation governing linearized displacements in a uniform elastic medium of density and Lamé constants and . Derive solutions for monochromatic plane and waves, and find the corresponding wave speeds and .
Such an elastic solid occupies the half-space , and the boundary is clamped rigidly so that . A plane -wave with frequency and wavenumber is incident on the boundary. At some angles of incidence, there results both a reflected -wave with frequency and wavenumber and a reflected -wave with frequency and wavenumber . Relate the frequencies and wavenumbers of the reflected waves to those of the incident wave. At what angles of incidence will there be a reflected -wave?
Find the amplitudes of the reflected waves as multiples of the amplitude of the incident wave. Confirm that these amplitudes give the sum of the time-averaged vertical fluxes of energy of the reflected waves equal to the time-averaged vertical flux of energy of the incident wave.
[Results concerning the energy flux, energy density and kinetic energy density in a plane elastic wave may be quoted without proof.]